Wiki Book Mounir Gmati Wiki Book Mounir Gmati Lévitation Quantique: Les ingrédients et la recette

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Contents

Articles

Superconductivity 1 1 Type-I superconductor 14 Yttrium barium copper oxide 15 nitrogen 20

Magnetism 23 23 33 Flux pinning 54 Magnetic levitation 55 References Article Sources and Contributors 63 Image Sources, Licenses and Contributors 65 Article Licenses License 66 Wiki Book Mounir Gmati 1

Superconductivity

Superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by on April 8, 1911 in Leiden. Like and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It is characterized by the Meissner effect, the complete ejection of magnetic field lines from the interior of the superconductor as it transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the A levitating above a high-temperature idealization of perfect conductivity in classical physics. superconductor, cooled with liquid nitrogen. Persistent electric current flows on the surface of The electrical resistivity of a metallic conductor decreases gradually as the superconductor, acting to exclude the temperature is lowered. In ordinary conductors, such as copper or magnetic field of the magnet (Faraday's law of silver, this decrease is limited by impurities and other defects. Even induction). This current effectively forms an electromagnet that repels the magnet. near absolute zero, a real sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. An electric current flowing in a loop of superconducting wire can persist indefinitely with no power source.[1]

In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 90 K (−183 °C). Such a high transition temperature is theoretically impossible for a conventional superconductor, leading the materials to be termed high-temperature superconductors. Liquid nitrogen boils at 77 K, A high-temperature superconductor levitating facilitating many experiments and applications that are less practical at above a magnet lower temperatures. In conventional superconductors, are held together in pairs by an attraction mediated by lattice . The best available model of high-temperature superconductivity is still somewhat crude. There is a hypothesis that pairing in high-temperature superconductors is mediated by short-range spin waves known as paramagnons.

Classification

There is not just one criterion to classify superconductors. The most common are • By their physical properties: they can be Type I (if their transition is of first order) or Type II (if their is of second order). • By the theory to explain them: they can be conventional (if they are explained by the BCS theory or its derivatives) or unconventional (if not). • By their critical temperature: they can be high temperature (generally considered if they reach the superconducting state just cooling them with liquid nitrogen, that is, if T > 77 K), or low temperature (generally c if they need other techniques to be cooled under their critical temperature). Superconductivity Wiki Book Mounir Gmati 2

• By material: they can be chemical elements (as mercury or lead), alloys (as niobium-titanium or germanium-niobium or niobium nitride), ceramics (as YBCO or the magnesium diboride), or organic superconductors (as fullerenes or carbon nanotubes, though these examples technically might be included among the chemical elements as they are composed entirely of carbon).

Elementary properties of superconductors Most of the physical properties of superconductors vary from material to material, such as the heat capacity and the critical temperature, critical field, and critical current density at which superconductivity is destroyed. On the other hand, there is a class of properties that are independent of the underlying material. For instance, all superconductors have exactly zero resistivity to low applied currents when there is no magnetic field present or if the applied field does not exceed a critical value. The existence of these "universal" properties implies that superconductivity is a thermodynamic phase, and thus possesses certain distinguishing properties which are largely independent of microscopic details.

Zero electrical DC resistance

The simplest method to measure the electrical resistance of a sample of some material is to place it in an electrical circuit in series with a current source I and measure the resulting voltage V across the sample. The resistance of the sample is given by Ohm's law as R = V/I. If the voltage is zero, this means that the resistance is zero. Superconductors are also able to maintain a current with no applied voltage whatsoever, a property exploited in superconducting electromagnets such as those found in MRI machines. Experiments have demonstrated that currents in superconducting coils can persist Electric cables for accelerators at CERN: top, for years without any measurable degradation. Experimental evidence regular cables for LEP; bottom, superconducting points to a current lifetime of at least 100,000 years. Theoretical cables for the LHC estimates for the lifetime of a persistent current can exceed the estimated lifetime of the universe, depending on the wire geometry and the temperature.[1]

In a normal conductor, an electric current may be visualized as a fluid of electrons moving across a heavy ionic lattice. The electrons are constantly colliding with the ions in the lattice, and during each collision some of the energy carried by the current is absorbed by the lattice and converted into heat, which is essentially the vibrational kinetic energy of the lattice ions. As a result, the energy carried by the current is constantly being dissipated. This is the phenomenon of electrical resistance. The situation is different in a superconductor. In a conventional superconductor, the electronic fluid cannot be resolved into individual electrons. Instead, it consists of bound pairs of electrons known as Cooper pairs. This pairing is caused by an attractive force between electrons from the exchange of phonons. Due to quantum mechanics, the energy spectrum of this fluid possesses an energy gap, meaning there is a minimum amount of energy ΔE that must be supplied in order to excite the fluid. Therefore, if ΔE is larger than the thermal energy of the lattice, given by kT, where k is Boltzmann's constant and T is the temperature, the fluid will not be scattered by the lattice. The Cooper pair fluid is thus a superfluid, meaning it can flow without energy dissipation. In a class of superconductors known as type II superconductors, including all known high-temperature superconductors, an extremely small amount of resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of vortices in the electronic superfluid, which dissipates some of the energy carried by the current. If the current is sufficiently small, the vortices are stationary, Superconductivity Wiki Book Mounir Gmati 3

and the resistivity vanishes. The resistance due to this effect is tiny compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen into a disordered but stationary phase known as a "vortex glass". Below this vortex glass transition temperature, the resistance of the material becomes truly zero.

Superconducting phase transition

In superconducting materials, the characteristics of superconductivity appear when the temperature T is lowered below a critical temperature T . The value of this critical c temperature varies from material to material. Conventional superconductors usually have critical temperatures ranging from around 20 K to less than 1 K. mercury, for example, has a critical temperature of 4.2 K. As of 2009, the highest critical temperature found for a conventional superconductor is 39 K [2] [3] for magnesium diboride (MgB ), Behavior of heat capacity (c , blue) and resistivity (ρ, green) at the superconducting phase 2 v although this material displays enough transition exotic properties that there is some doubt about classifying it as a "conventional" superconductor.[4] Cuprate superconductors can have much higher critical temperatures: YBa Cu O , one of the first cuprate superconductors to be discovered, has a critical 2 3 7 temperature of 92 K, and mercury-based cuprates have been found with critical temperatures in excess of 130 K. The explanation for these high critical temperatures remains unknown. Electron pairing due to exchanges explains superconductivity in conventional superconductors, but it does not explain superconductivity in the newer superconductors that have a very high critical temperature.

Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external magnetic field is applied which is greater than the critical magnetic field. This is because the Gibbs free energy of the superconducting phase increases quadratically with the magnetic field while the free energy of the normal phase is roughly independent of the magnetic field. If the material superconducts in the absence of a field, then the superconducting phase free energy is lower than that of the normal phase and so for some finite value of the magnetic field (proportional to the square root of the difference of the free energies at zero magnetic field) the two free energies will be equal and a phase transition to the normal phase will occur. More generally, a higher temperature and a stronger magnetic field lead to a smaller fraction of the electrons in the superconducting band and consequently a longer London penetration depth of external magnetic fields and currents. The penetration depth becomes infinite at the phase transition. The onset of superconductivity is accompanied by abrupt changes in various physical properties, which is the hallmark of a phase transition. For example, the electronic heat capacity is proportional to the temperature in the normal (non-superconducting) regime. At the superconducting transition, it suffers a discontinuous jump and thereafter ceases to be linear. At low temperatures, it varies instead as e−α /T for some constant, α. This exponential behavior is one of the pieces of evidence for the existence of the energy gap. Superconductivity Wiki Book Mounir Gmati 4

The order of the superconducting phase transition was long a matter of debate. Experiments indicate that the transition is second-order, meaning there is no latent heat. However in the presence of an external magnetic field there is latent heat, as a result of the fact that the superconducting phase has a lower entropy below the critical temperature than the normal phase. It has been experimentally demonstrated[5] that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material. Calculations in the 1970s suggested that it may actually be weakly first-order due to the effect of long-range fluctuations in the electromagnetic field. In the 1980s it was shown theoretically with the help of a disorder field theory, in which the vortex lines of the superconductor play a major role, that the transition is of second order within the type II regime and of first order (i.e., latent heat) within the type I regime, and that the two regions are separated by a tricritical point.[6] The results were confirmed by Monte Carlo computer simulations.[7]

Meissner effect When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. The Meissner effect does not cause the field to be completely ejected but instead the field penetrates the superconductor but only to a very small distance, characterized by a parameter λ, called the London penetration depth, decaying exponentially to zero within the bulk of the material. The Meissner effect is a defining characteristic of superconductivity. For most superconductors, the London penetration depth is on the order of 100 nm. The Meissner effect is sometimes confused with the kind of one would expect in a perfect : according to Lenz's law, when a changing magnetic field is applied to a conductor, it will induce an electric current in the conductor that creates an opposing magnetic field. In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic field exactly cancels the applied field. The Meissner effect is distinct from this—it is the spontaneous expulsion which occurs during transition to superconductivity. Suppose we have a material in its normal state, containing a constant internal magnetic field. When the material is cooled below the critical temperature, we would observe the abrupt expulsion of the internal magnetic field, which we would not expect based on Lenz's law. The Meissner effect was given a phenomenological explanation by the brothers Fritz and Heinz London, who showed that the electromagnetic free energy in a superconductor is minimized provided

where H is the magnetic field and λ is the London penetration depth. This equation, which is known as the London equation, predicts that the magnetic field in a superconductor decays exponentially from whatever value it possesses at the surface. A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H . Depending on the geometry of the sample, one may obtain an c intermediate state[8] consisting of a baroque pattern[9] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H leads to a mixed state (also known as the vortex state) in which an increasing amount of c1 magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H , superconductivity is destroyed. The mixed state is c2 actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium, technetium, vanadium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II. Superconductivity Wiki Book Mounir Gmati 5

London moment Conversely, a spinning superconductor generates a magnetic field, precisely aligned with the spin axis. The effect, the London moment, was put to good use in Gravity Probe B. This experiment measured the magnetic fields of four superconducting gyroscopes to determine their spin axes. This was critical to the experiment since it is one of the few ways to accurately determine the spin axis of an otherwise featureless sphere.

Theories of superconductivity Since the discovery of superconductivity, great efforts have been devoted to finding out how and why it works. During the 1950s, theoretical condensed matter physicists arrived at a solid understanding of "conventional" superconductivity, through a pair of remarkable and important theories: the phenomenological Ginzburg-Landau theory (1950) and the microscopic BCS theory (1957).[10] [11] Generalizations of these theories form the basis for understanding the closely related phenomenon of , because they fall into the Lambda transition universality class, but the extent to which similar generalizations can be applied to unconventional superconductors as well is still controversial. The four-dimensional extension of the Ginzburg-Landau theory, the Coleman-Weinberg model, is important in quantum field theory and cosmology.

London theory The first phenomenological theory of superconductivity was London theory. It was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. A major triumph of the equations of this theory is their ability to explain the Meissner effect[12] , wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold. By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface[13] . There are two London equations:

The first equation follows from the Newton's second law for superconducting electrons.

History of superconductivity Superconductivity was discovered on April 8, 1911 by Heike Kamerlingh Onnes, who was studying the resistance of solid mercury at cryogenic temperatures using the recently-produced liquid helium as a refrigerant. At the temperature of 4.2 K, he observed that the resistance abruptly disappeared.[14] In the same experiment, he also observed the superfluid transition of helium at 2.2 K, without recognizing its significance. (The precise date and circumstances of the discovery were only reconstructed a century later, when Onnes's notebook was found.)[15] In subsequent decades, superconductivity was observed in several other materials. In 1913, lead was found to superconduct at 7 K, and in 1941 niobium nitride was found to superconduct at 16 K. The next important step in understanding superconductivity occurred in 1933, when Meissner and Ochsenfeld discovered that superconductors expelled applied magnetic fields, a phenomenon which has come to be known as the Meissner effect.[16] In 1935, F. and H. London showed that the Meissner effect was a consequence of the minimization of the electromagnetic free energy carried by superconducting current.[17] In 1950, the phenomenological Ginzburg-Landau theory of superconductivity was devised by Landau and Ginzburg.[18] This theory, which combined Landau's theory of second-order phase transitions with a Schrödinger-like wave equation, had great success in explaining the macroscopic properties of superconductors. In particular, Abrikosov showed that Ginzburg-Landau theory predicts the division of superconductors into the two categories now referred to as Type I and Type II. Abrikosov and Ginzburg were awarded the 2003 Nobel Prize for Superconductivity Wiki Book Mounir Gmati 6

their work (Landau had received the 1962 Nobel Prize for other work, and died in 1968). Also in 1950, Maxwell and Reynolds et al. found that the critical temperature of a superconductor depends on the isotopic mass of the constituent element.[19] [20] This important discovery pointed to the electron-phonon interaction as the microscopic mechanism responsible for superconductivity. The complete microscopic theory of superconductivity was finally proposed in 1957 by Bardeen, Cooper and Schrieffer.[11] Independently, the superconductivity phenomenon was explained by Nikolay Bogolyubov. This BCS theory explained the superconducting current as a superfluid of Cooper pairs, pairs of electrons interacting through the exchange of phonons. For this work, the authors were awarded the Nobel Prize in 1972. The BCS theory was set on a firmer footing in 1958, when Bogolyubov showed that the BCS wavefunction, which had originally been derived from a variational argument, could be obtained using a canonical transformation of the electronic Hamiltonian.[21] In 1959, Lev Gor'kov showed that the BCS theory reduced to the Ginzburg-Landau theory close to the critical temperature.[22] In 1962, the first commercial superconducting wire, a niobium-titanium alloy, was developed by researchers at Westinghouse, allowing the construction of the first practical superconducting . In the same year, Josephson made the important theoretical prediction that a supercurrent can flow between two pieces of superconductor separated by a thin layer of .[23] This phenomenon, now called the , is exploited by superconducting devices such as SQUIDs. It is used in the most accurate available measurements of the magnetic flux quantum , and thus (coupled with the quantum Hall resistivity) for Planck's constant h. Josephson

was awarded the Nobel Prize for this work in 1973. In 2008, it was discovered that the same mechanism that produces superconductivity could produce a superinsulator state in some materials, with almost infinite electrical resistance.[24]

High-temperature superconductivity Until 1986, physicists had believed that BCS theory forbade superconductivity at temperatures above about 30 K. In that year, Bednorz and Müller discovered superconductivity in a lanthanum-based cuprate perovskite material, which had a transition temperature of 35 K (Nobel Prize in Physics, 1987).[25] It was soon found that replacing the lanthanum with yttrium (i.e., making YBCO) raised the critical temperature to 92 K, which was important because liquid nitrogen could then be used as a refrigerant (the boiling point of nitrogen is 77 K at atmospheric pressure).[26] This is important commercially because liquid nitrogen can be produced cheaply on-site from air, and is not prone to some of the problems (for instance solid air plugs) of helium in piping. Many other cuprate superconductors have since been discovered, and the theory of superconductivity in these materials is one of the major outstanding challenges of theoretical .[27] From about 1993, the highest temperature superconductor was a ceramic material consisting of thallium, mercury, copper, barium, calcium and oxygen (HgBa Ca Cu O ) with T = 138 K.[28] 2 2 3 8+δ c In February 2008, an iron-based family of high-temperature superconductors was discovered.[29] [30] Hideo Hosono, of the Tokyo Institute of Technology, and colleagues found lanthanum oxygen fluorine iron arsenide (LaO F FeAs), an oxypnictide that superconducts below 26 K. Replacing the lanthanum in LaO F FeAs with 1-x x 1−x x samarium leads to superconductors that work at 55 K.[31] Superconductivity Wiki Book Mounir Gmati 7

Crystal structure of high-temperature ceramic superconductors The structure of a high-T superconductor is closely related to perovskite structure, and the structure of these c compounds has been described as a distorted, oxygen deficient multi-layered perovskite structure. One of the properties of the crystal structure of oxide superconductors is an alternating multi-layer of CuO planes with 2 superconductivity taking place between these layers. The more layers of CuO the higher T . This structure causes a 2 c large anisotropy in normal conducting and superconducting properties, since electrical currents are carried by holes induced in the oxygen sites of the CuO sheets. The electrical conduction is highly anisotropic, with a much higher 2 conductivity parallel to the CuO plane than in the perpendicular direction. Generally, Critical temperatures depend 2 on the chemical compositions, cations substitutions and oxygen content. They can be classified as superstripes; i.e., particular realizations of superlattices at atomic limit made of superconducting atomic layers, wires, dots separated by spacer layers, that gives multiband and multigap superconductivity.

YBaCuO superconductors

The first superconductor found with T > 77 K (liquid nitrogen boiling c point) is yttrium barium copper oxide (YBa Cu O ), the proportions 2 3 7-x of the 3 different metals in the YBa Cu O superconductor are in the 2 3 7 mole ratio of 1 to 2 to 3 for yttrium to barium to copper respectively. Thus, this particular superconductor is often referred to as the 123 superconductor.

The unit cell of YBa Cu O consists of three pseudocubic elementary 2 3 7 perovskite unit cells. Each perovskite unit cell contains a Y or Ba atom at the center: Ba in the bottom unit cell, Y in the middle one, and Ba in the top unit cell. Thus, Y and Ba are stacked in the sequence [Ba–Y–Ba] along the c-axis. All corner sites of the unit cell are occupied by Cu, which has two different coordinations, Cu(1) and Cu(2), with respect to oxygen. There are four possible crystallographic sites for oxygen: O(1), O(2), O(3) and O(4).[32] The coordination polyhedra of Y and Ba with respect to oxygen are different. The tripling of the perovskite unit cell leads to nine oxygen atoms, whereas YBCO unit cell YBa Cu O has seven oxygen atoms and, therefore, is referred to as an 2 3 7 oxygen-deficient perovskite structure. The structure has a stacking of different layers: (CuO)(BaO)(CuO )(Y)(CuO )(BaO)(CuO). One of the key feature of the unit cell of YBa Cu O 2 2 2 3 7-x (YBCO) is the presence of two layers of CuO . The role of the Y plane is to serve as a spacer between two CuO 2 2 planes. In YBCO, the Cu–O chains are known to play an important role for superconductivity. T is maximal near 92 c K when x ≈ 0.15 and the structure is orthorhombic. Superconductivity disappears at x ≈ 0.6, where the structural transformation of YBCO occurs from orthorhombic to tetragonal.[33] Superconductivity Wiki Book Mounir Gmati 8

Bi-, Tl- and Hg-based high-T superconductors c The crystal structure of Bi-, Tl- and Hg-based high-T superconductors are very similar. Like YBCO, the c perovskite-type feature and the presence of CuO layers also exist in these superconductors. However, unlike YBCO, 2 Cu–O chains are not present in these superconductors. The YBCO superconductor has an orthorhombic structure, whereas the other high-T superconductors have a tetragonal structure. c The Bi–Sr–Ca–Cu–O system has three superconducting phases forming a homologous series as Bi Sr Ca Cu O (n = 1, 2 and 3). These three phases are Bi-2201, Bi-2212 and Bi-2223, having transition 2 2 n−1 n 4+2n+x temperatures of 20, 85 and 110 K, respectively, where the numbering system represent number of atoms for Bi, Sr, Ca and Cu respectively.[34] The two phases have a tetragonal structure which consists of two sheared crystallographic unit cells. The unit cell of these phases has double Bi–O planes which are stacked in a way that the Bi atom of one plane sits below the oxygen atom of the next consecutive plane. The Ca atom forms a layer within the interior of the CuO layers in both Bi-2212 and Bi-2223; there is no Ca layer in the Bi-2201 phase. The three phases 2 differ with each other in the number of CuO planes; Bi-2201, Bi-2212 and Bi-2223 phases have one, two and three 2 CuO planes, respectively. The c axis of these phases increases with the number of CuO planes (see table below). 2 2 The coordination of the Cu atom is different in the three phases. The Cu atom forms an octahedral coordination with respect to oxygen atoms in the 2201 phase, whereas in 2212, the Cu atom is surrounded by five oxygen atoms in a pyramidal arrangement. In the 2223 structure, Cu has two coordinations with respect to oxygen: one Cu atom is bonded with four oxygen atoms in square planar configuration and another Cu atom is coordinated with five oxygen atoms in a pyramidal arrangement.[35] Tl–Ba–Ca–Cu–O superconductor: The first series of the Tl-based superconductor containing one Tl–O layer has the general formula TlBa Ca Cu O ,[36] whereas the second series containing two Tl–O layers has a formula of 2 n-1 n 2n+3 Tl Ba Ca Cu O with n = 1, 2 and 3. In the structure of Tl Ba CuO (Tl-2201), there is one CuO layer with 2 2 n-1 n 2n+4 2 2 6 2 the stacking sequence (Tl–O) (Tl–O) (Ba–O) (Cu–O) (Ba–O) (Tl–O) (Tl–O). In Tl Ba CaCu O (Tl-2212), there 2 2 2 8 are two Cu–O layers with a Ca layer in between. Similar to the Tl Ba CuO structure, Tl–O layers are present 2 2 6 outside the Ba–O layers. In Tl Ba Ca Cu O (Tl-2223), there are three CuO2 layers enclosing Ca layers between 2 2 2 3 10 each of these. In Tl-based superconductors, T is found to increase with the increase in CuO layers. However, the c 2 value of T decreases after four CuO layers in TlBa Ca Cu O , and in the Tl Ba Ca Cu O compound, it c 2 2 n-1 n 2n+3 2 2 n-1 n 2n+4 decreases after three CuO layers.[37] 2 Hg–Ba–Ca–Cu–O superconductor: The crystal structure of HgBa CuO (Hg-1201),[38] HgBa CaCu O 2 4 2 2 6 (Hg-1212) and HgBa Ca Cu O (Hg-1223) is similar to that of Tl-1201, Tl-1212 and Tl-1223, with Hg in place 2 2 3 8 of Tl. It is noteworthy that the T of the Hg compound (Hg-1201) containing one CuO layer is much larger as c 2 compared to the one-CuO -layer compound of thallium (Tl-1201). In the Hg-based superconductor, T is also found 2 c to increase as the CuO layer increases. For Hg-1201, Hg-1212 and Hg-1223, the values of T are 94, 128 and 134 K 2 c respectively, as shown in table below. The observation that the T of Hg-1223 increases to 153 K under high pressure c indicates that the T of this compound is very sensitive to the structure of the compound.[39] c Superconductivity Wiki Book Mounir Gmati 9

Critical temperature (T ), crystal structure and lattice constants of some high-T c c superconductors

Formula Notation T (K) No. of Cu-O Crystal structure c planes in unit cell

YBa Cu O 123 92 2 Orthorhombic 2 3 7 Bi Sr CuO Bi-2201 20 1 Tetragonal 2 2 6 Bi Sr CaCu O Bi-2212 85 2 Tetragonal 2 2 2 8 Bi Sr Ca Cu O Bi-2223 110 3 Tetragonal 2 2 2 3 6 Tl Ba CuO Tl-2201 80 1 Tetragonal 2 2 6 Tl Ba CaCu O Tl-2212 108 2 Tetragonal 2 2 2 8 Tl Ba Ca Cu O Tl-2223 125 3 Tetragonal 2 2 2 3 10 TlBa Ca Cu O Tl-1234 122 4 Tetragonal 2 3 4 11 HgBa CuO Hg-1201 94 1 Tetragonal 2 4 HgBa CaCu O Hg-1212 128 2 Tetragonal 2 2 6 HgBa Ca Cu O Hg-1223 134 3 Tetragonal 2 2 3 8

Preparation of high-T superconductors c The simplest method for preparing high-T superconductors is a c solid-state thermochemical reaction involving mixing, calcination and sintering. The appropriate amounts of precursor powders, usually oxides and carbonates, are mixed thoroughly using a ball mill. Solution chemistry processes such as coprecipitation, freeze-drying and sol-gel methods are alternative ways for preparing a homogenous mixture. These powders are calcined in the temperature range from 800 °C to 950 °C for several hours. The powders are cooled, reground and

calcined again. This process is repeated several times to get Superconductor timeline homogenous material. The powders are subsequently compacted to pellets and sintered. The sintering environment such as temperature, annealing time, atmosphere and cooling rate play a very important role in getting good high-T superconducting materials. The YBa Cu O compound is c 2 3 7-x prepared by calcination and sintering of a homogenous mixture of Y O , BaCO and CuO in the appropriate atomic 2 3 3 ratio. Calcination is done at 900–950 °C, whereas sintering is done at 950 °C in an oxygen atmosphere. The oxygen stoichiometry in this material is very crucial for obtaining a superconducting YBa Cu O compound. At the time 2 3 7−x of sintering, the semiconducting tetragonal YBa Cu O compound is formed, which, on slow cooling in oxygen 2 3 6 atmosphere, turns into superconducting YBa Cu O . The uptake and loss of oxygen are reversible in 2 3 7−x YBa Cu O . A fully oxidized orthorhombic YBa Cu O sample can be transformed into tetragonal YBa Cu O 2 3 7−x 2 3 7−x 2 3 6 by heating in a vacuum at temperature above 700 °C.[33]

The preparation of Bi-, Tl- and Hg-based high-T superconductors is difficult compared to YBCO. Problems in these c superconductors arise because of the existence of three or more phases having a similar layered structure. Thus, syntactic intergrowth and defects such as stacking faults occur during synthesis and it becomes difficult to isolate a single superconducting phase. For Bi–Sr–Ca–Cu–O, it is relatively simple to prepare the Bi-2212 (T ≈ 85 K) phase, c whereas it is very difficult to prepare a single phase of Bi-2223 (T ≈ 110 K). The Bi-2212 phase appears only after c few hours of sintering at 860–870 °C, but the larger fraction of the Bi-2223 phase is formed after a long reaction Superconductivity Wiki Book Mounir Gmati 10

time of more than a week at 870 °C.[35] Although the substitution of Pb in the Bi–Sr–Ca–Cu–O compound has been found to promote the growth of the high-T phase,[40] a long sintering time is still required. c

Possible superconductivity of the vacuum Maxim Chernodub of the French National Centre for Scientific Research has postulated that the vacuum can become a superconductor in magnetic fields of 1016 Tesla or more,[41] at temperatures of at least a billion,[42] perhaps billions of degrees.[43]

Applications Superconducting magnets are some of the most powerful electromagnets known. They are used in MRI/NMR machines, mass spectrometers, and the beam-steering magnets used in particle accelerators. They can also be used for magnetic separation, where weakly magnetic particles are extracted from a background of less or non-magnetic particles, as in the pigment industries. In the 1950s and 1960s, superconductors were used to build experimental digital computers using cryotron switches. More recently, superconductors have been used to make digital circuits based on rapid single flux quantum technology and RF and microwave filters for mobile phone base stations. Superconductors are used to build Josephson junctions which are the building blocks of SQUIDs (superconducting quantum interference devices), the most sensitive magnetometers known. SQUIDs are used in scanning SQUID microscopes and magnetoencephalography. Series of Josephson devices are used to realize the SI volt. Depending on the particular mode of operation, a superconductor-insulator-superconductor Josephson junction can be used as a photon detector or as a mixer. The large resistance change at the transition from the normal- to the superconducting state is used to build thermometers in cryogenic micro-calorimeter photon detectors. The same effect is used in ultrasensitive bolometers made from superconducting materials. Other early markets are arising where the relative efficiency, size and weight advantages of devices based on high-temperature superconductivity outweigh the additional costs involved. Promising future applications include high-performance smart grid, electric power transmission, transformers, power storage devices, electric motors (e.g. for vehicle propulsion, as in vactrains or maglev trains), magnetic levitation devices, fault current limiters, nanoscopic materials such as buckyballs, nanotubes, composite materials and superconducting magnetic refrigeration. However, superconductivity is sensitive to moving magnetic fields so applications that use alternating current (e.g. transformers) will be more difficult to develop than those that rely upon direct current.

Nobel Prize for Superconductivity 1913 Heike Kamerlingh Onnes on Matter at low temperature 1972 , Leon N. Cooper, J. Robert Schrieffer on Theory of superconductivity 1973 , , Brian D. Josephson on Tunneling in superconductors 1987 , Alex K. Müller on High-temperature superconductivity 2003 Alexei A. Abrikosov, Vitaly L. Ginzburg, Anthony J. Leggett on Pioneering contributions to the theory of superconductors and superfluids. [44] Superconductivity Wiki Book Mounir Gmati 11

References

[1] John C. Gallop (1990). SQUIDS, the Josephson Effects and Superconducting Electronics (http:/ / books. google. com/ ?id=ad8_JsfCdKQC& printsec=frontcover). CRC Press. pp. 3, 20. ISBN 0750300515. . [2] Jun Nagamatsu, Norimasa Nakagawa, Takahiro Muranaka, Yuji Zenitani and Jun Akimitsu (2001). "Superconductivity at 39 K in magnesium diboride". Nature 410 (6824): 63. Bibcode 2001Natur.410...63N. doi:10.1038/35065039. PMID 11242039. [3] Paul Preuss (14 August 2002). "A most unusual superconductor and how it works: first-principles calculation explains the strange behavior of

magnesium diboride" (http:/ / www. lbl. gov/ Science-Articles/ Archive/ MSD-superconductor-Cohen-Louie. html). Research News (Lawrence Berkeley National Laboratory). . Retrieved 2009-10-28.

[4] Hamish Johnston (17 February 2009). "Type-1.5 superconductor shows its stripes" (http:/ / physicsworld. com/ cws/ article/ news/ 37806). Physics World (Institute of Physics). . Retrieved 2009-10-28. [5] R. L. Dolecek (1954). "Adiabatic of a Superconducting Sphere". Physical Review 96 (1): 25–28. Bibcode 1954PhRv...96...25D. doi:10.1103/PhysRev.96.25.

[6] H. Kleinert (1982). "Disorder Version of the Abelian Higgs Model and the Order of the Superconductive Phase Transition" (http:/ / www.

physik. fu-berlin. de/ ~kleinert/ 97/ 97. pdf). Lettere al Nuovo Cimento 35 (13): 405–412. doi:10.1007/BF02754760. .

[7] J. Hove, S. Mo, A. Sudbo (2002). "Vortex interactions and thermally induced crossover from type-I to type-II superconductivity" (http:/ /

www. physik. fu-berlin. de/ ~kleinert/ papers/ sudbotre064524. pdf). Physical Review B 66 (6): 064524. arXiv:cond-mat/0202215. Bibcode 2002PhRvB..66f4524H. doi:10.1103/PhysRevB.66.064524. . [8] Lev D. Landau, Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course of Theoretical Physics. 8. Oxford: Butterworth-Heinemann. ISBN 0-7506-2634-8. [9] David J. E. Callaway (1990). "On the remarkable structure of the superconducting intermediate state". Nuclear Physics B 344 (3): 627–645. Bibcode 1990NuPhB.344..627C. doi:10.1016/0550-3213(90)90672-Z. [10] J. Bardeen, L. N. Cooper and J. R. Schrieffer (1957). "Microscopic Theory of Superconductivity". Physical Review 106 (1): 162–164. Bibcode 1957PhRv..106..162B. doi:10.1103/PhysRev.106.162. [11] J. Bardeen, L. N. Cooper and J. R. Schrieffer (1957). "Theory of Superconductivity". Physical Review 108 (5): 1175–1205. Bibcode 1957PhRv..108.1175B. doi:10.1103/PhysRev.108.1175. [12] Meissner, W.; R. Ochsenfeld (1933). "Ein neuer Effekt bei Eintritt der Supraleitfähigkeit". Naturwissenschaften 21 (44): 787. Bibcode 1933NW.....21..787M. doi:10.1007/BF01504252.

[13] "The London equations" (http:/ / openlearn. open. ac. uk/ mod/ oucontent/ view. php?id=398540& section=3. 3). The Open University. . Retrieved 2011-10-16. [14] H. K. Onnes (1911). "The resistance of pure mercury at helium temperatures". Commun. Phys. Lab. Univ. Leiden 12: 120.

[15] The Discovery of Superconductivity (http:/ / ilorentz. org/ history/ cold/ DelftKes_HKO_PT. pdf) [16] W. Meissner and R. Ochsenfeld (1933). "Ein neuer Effekt bei Eintritt der Supraleitfähigkeit". Naturwissenschaften 21 (44): 787–788. Bibcode 1933NW.....21..787M. doi:10.1007/BF01504252. [17] F. London and H. London (1935). "The Electromagnetic Equations of the Supraconductor". Proceedings of the Royal Society of London A 149 (866): 71–88. Bibcode 1935RSPSA.149...71L. doi:10.1098/rspa.1935.0048. JSTOR 96265. [18] V. L. Ginzburg and L.D. Landau (1950). "On the theory of superconductivity". Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 20: 1064. [19] E. Maxwell (1950). "Isotope Effect in the Superconductivity of Mercury". Physical Review 78 (4): 477. Bibcode 1950PhRv...78..477M. doi:10.1103/PhysRev.78.477. [20] C. A. Reynolds, B. Serin, W. H. Wright and L. B. Nesbitt (1950). "Superconductivity of Isotopes of Mercury". Physical Review 78 (4): 487. Bibcode 1950PhRv...78..487R. doi:10.1103/PhysRev.78.487. [21] N. N. Bogoliubov (1958). "A new method in the theory of superconductivity". Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 34: 58. [22] L. P. Gor'kov (1959). "Microscopic derivation of the Ginzburg--Landau equations in the theory of superconductivity". Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 36: 1364. [23] B. D. Josephson (1962). "Possible new effects in superconductive tunnelling". Physics Letters 1 (7): 251–253. Bibcode 1962PhL.....1..251J. doi:10.1016/0031-9163(62)91369-0.

[24] "Newly discovered fundamental , a superinsulator, has been created." (http:/ / www. sciencedaily. com/ releases/ 2008/ 04/

080408160614. htm). Science Daily. April 9, 2008. . Retrieved 2008-10-23. [25] J. G. Bednorz and K. A. Mueller (1986). "Possible high T superconductivity in the Ba-La-Cu-O system". Zeitschrift für Physik B 64 (2): C 189–193. Bibcode 1986ZPhyB..64..189B. doi:10.1007/BF01303701. [26] M. K. Wu et al. (1987). "Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O Compound System at Ambient Pressure". Physical Review Letters 58 (9): 908–910. Bibcode 1987PhRvL..58..908W. doi:10.1103/PhysRevLett.58.908. PMID 10035069.

[27] Alexei A. Abrikosov (8 December 2003). "type II Superconductors and the Vortex Lattice" (http:/ / nobelprize. org/ nobel_prizes/ physics/

laureates/ 2003/ abrikosov-lecture. html). Nobel Lecture. . [28] P. Dai, B. C. Chakoumakos, G. F. Sun, K. W. Wong, Y. Xin and D. F. Lu (1995). "Synthesis and neutron powder diffraction study of the superconductor HgBa Ca Cu O by Tl substitution". Physica C 243 (3–4): 201–206. Bibcode 1995PhyC..243..201D. 2 2 3 8+δ doi:10.1016/0921-4534(94)02461-8. [29] Hiroki Takahashi, Kazumi Igawa, Kazunobu Arii, Yoichi Kamihara, Masahiro Hirano, Hideo Hosono (2008). "Superconductivity at 43 K in an iron-based layered compound LaO F FeAs". Nature 453 (7193): 376–378. Bibcode 2008Natur.453..376T. doi:10.1038/nature06972. 1−x x Superconductivity Wiki Book Mounir Gmati 12

PMID 18432191.

[30] Adrian Cho. "Second Family of High-Temperature Superconductors Discovered" (http:/ / sciencenow. sciencemag. org/ cgi/ content/ full/

2008/ 417/ 1). ScienceNOW Daily News. . [31] Zhi-An Ren et al. (2008). "Superconductivity and phase diagram in iron-based arsenic-oxides ReFeAsO1-d (Re = rare-earth metal) without fluorine doping". EPL 83: 17002. Bibcode 2008EL.....8317002R. doi:10.1209/0295-5075/83/17002. [32] R. Hazen et al. (1987). "Crystallographic description of phases in the Y-Ba-Cu-O superconductor". Physical Review B 35 (13): 7238. Bibcode 1987PhRvB..35.7238H. doi:10.1103/PhysRevB.35.7238.

[33] Neeraj Khare (2003-05-01). Handbook of High-Temperature Superconductor Electronics (http:/ / books. google. com/ books?id=is-L_R2H3IAC). ISBN 0-8247-0823-7. . [34] R. Hazen et al. (1988). "Superconductivity in the high-T Bi-Ca-Sr-Cu-O system: Phase identification". Physical Review Letters 60 (12): c 1174. Bibcode 1988PhRvL..60.1174H. doi:10.1103/PhysRevLett.60.1174. [35] J. Tarascon et al. (1988). "Preparation, structure, and properties of the superconducting compound series Bi Sr Ca Cu O with n = 1, 2, 2 2 n−1 n y and 3". Physical Review B 38 (13): 8885. Bibcode 1988PhRvB..38.8885T. doi:10.1103/PhysRevB.38.8885. [36] Z. Sheng et al. (1988). "Superconductivity at 90 K in the Tl-Ba-Cu-O system". Physical Review Letters 60 (10): 937–940. Bibcode 1988PhRvL..60..937S. doi:10.1103/PhysRevLett.60.937. PMID 10037895. [37] Z.Z. Sheng, A. M. Hermann, Z. Z.; Hermann, A. M. (1988). "Superconductivity in the rare-earth-free Tl-Ba-Cu-O system above liquid-nitrogen temperature". Nature 332 (6159): 55. Bibcode 1988Natur.332...55S. doi:10.1038/332055a0. [38] S. N. Putilin et al. (1993). "Superconductivity at 94 K in HgBa2Cu04+δ". Nature 362 (6417): 226. Bibcode 1993Natur.362..226P. doi:10.1038/362226a0. [39] C. W. Chu et al. (1993). "Superconductivity above 150 K in HgBa2Ca2Cu3O8+δ at high pressures". Nature 365 (6444): 323. Bibcode 1993Natur.365..323C. doi:10.1038/365323a0. [40] D. Shi et al. (1989). "Origin of enhanced growth of the 110 K superconducting phase by Pb doping in the Bi-Sr-Ca-Cu-O system". Applied Physics Letters 55 (7): 699. Bibcode 1989ApPhL..55..699S. doi:10.1063/1.101573.

[41] Maggie McKee: How to turn the vacuum into a superconductor (http:/ / webcache. googleusercontent. com/

search?q=cache:yuuQSPuvjdAJ:www. newscientist. com/ article/ mg21028073. 800-how-to-turn-the-vacuum-into-a-superconductor. html+

"maxim+ chernodub"+ "new+ scientist"& cd=4& hl=en& ct=clnk& gl=uk& source=www. google. co. uk). New Scientist, April 8, 2011, retrieved May 13, 2011

[42] Alasdair Wilkins: Ultra-hot superconductors could spontaneously form in the vacuum of space (http:/ / io9. com/ 5787141/ ultra+ hot-superconductors-could-spontaneously-form-in-the-vacuum-of-space). io9, March 30, 2011, retrieved May 13, 2011

[43] Jon Cartwright: Superconductivity from nowhere (http:/ / physicsworld. com/ cws/ article/ news/ 45558). Physicworld.com, March 29, 2011, retrieved May 13, 2011

[44] The Nobel Prize for Physics, 1901–2008 [=http:/ / math. ucr. edu/ home/ baez/ physics/ Administrivia/ nobel. html "The Nobel Prize for Physics,1901-2008"]. =.

Further reading

• Hagen Kleinert (1989). "Superflow and Vortex Lines" (http:/ / www. physik. fu-berlin. de/ ~kleinert/

kleiner_reb1/ contents1. html). Gauge Fields in Condensed Matter. 1. World Scientific. ISBN 9971-5-0210-0. • Anatoly Larkin; Andrei Varlamov (2005). Theory of Fluctuations in Superconductors. Oxford University Press. ISBN 0198528159. • A. G. Lebed (2008). The Physics of Organic Superconductors and Conductors. 110 (1rst ed.). Springer. ISBN 978-3-540-76667-4. • Jean Matricon, Georges Waysand, Charles Glashausser (2003). The Cold Wars: A History of Superconductivity. Rutgers University Press. ISBN 0813532957.

• "Physicist Discovers Exotic Superconductivity" (http:/ / www. sciencedaily. com/ releases/ 2006/ 08/

060817101658. htm). ScienceDaily. 17 August 2006. • Michael Tinkham (2004). Introduction to Superconductivity (2nd ed.). Dover Books. ISBN 0-486-43503-2. • Terry Orlando, Kevin Delin (1991). Foundations of Applied Superconductivity. Prentice Hall. ISBN 978-0-2011-8323-8. • Paul Tipler, Ralph Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0-7167-4345-0. Superconductivity Wiki Book Mounir Gmati 13

External links

• Everything about superconductivity: properties, research, applications with videos, animations, games (http:/ /

www. superconductivity. eu) • Video about Type I Superconductors: R=0/transition temperatures/ B is a state variable/ Meissner effect/ Energy

gap(Giaever)/ BCS model (http:/ / alfredleitner. com)

• Superconductivity: Current in a Cape and Thermal Tights. An introduction to the topic for non-scientists (http:/ /

www. magnet. fsu. edu/ education/ tutorials/ magnetacademy/ superconductivity101/ ) National High Magnetic Field Laboratory

• Introduction to superconductivity (http:/ / www. ornl. gov/ reports/ m/ ornlm3063r1/ pt1. html)

• Lectures on Superconductivity (series of videos, including interviews with leading experts) (http:/ / www. msm.

cam. ac. uk/ ascg/ lectures/ )

• Superconducting Niobium Cavities (http:/ / www. sns. gov/ partnerlabs/ jlab. shtml)

• Superconductivity in everyday life : Interactive exhibition (http:/ / www. superlife. info) • Videos for various types of superconducting levitations including trains and hoolahoops – also videos of Ohm's

law in a superconductor (http:/ / h0. web. u-psud. fr/ supraconductivite/ vulgaFilms. html)

• Video of the Meissner effect from the NJIT Mathclub (http:/ / web. njit. edu/ ~mathclub/ superconductor/ index. html)

• Superconductivity News Update (http:/ / www. superconductivitynewsupdate. com)

• Superconductor Week Newsletter – industry news, links, et cetera (http:/ / www. superconductorweek. com)

• Superconducting Magnetic Levitation (http:/ / www. maniacworld. com/ Superconducting-Magnetic-Levitation. html)

• National Superconducting Cyclotron Laboratory at Michigan State University (http:/ / www. nscl. msu. edu)

• High Temperature Superconducting and Cryogenics in RF applications (http:/ / www. suptech. com/

hts_crfe_tech. htm)

• CERN Superconductors Database (http:/ / sdb-server. cern. ch/ mediawiki/ index. php/ Main_Page)

• Magnetisation of High Temperature superconductors by the flux pumping method (http:/ / www. fluxpump. co.

uk/ default. aspx)

• YouTube Video Levitating magnet (http:/ / youtube. com/ watch?v=indyz6O-Xyw& feature=user)

• Isotope effect in superconductivity (http:/ / www. physics. csulb. edu/ ~abill/ isotope. html)

• International Workshop on superconductivity in Diamond and Related Materials (free download papers) (http:/ /

www. iop. org/ EJ/ toc/ 1468-6996/ 9/ 4) • New Diamond and Frontier Carbon Technology Volume 17, No.1 Special Issue on Superconductivity in CVD

Diamond (http:/ / www. nims. go. jp/ NFM/ NDFCT17/ NDFCT17. html)

• DoITPoMS Teaching and Learning Package – "Superconductivity" (http:/ / www. doitpoms. ac. uk/ tlplib/

superconductivity/ index. php)

• The Nobel Prize for Physics, 1901–2008 (http:/ / math. ucr. edu/ home/ baez/ physics/ Administrivia/ nobel. html) Type-I superconductorWiki Book Mounir Gmati 14 Type-I superconductor

Superconductors cannot be penetrated by magnetic flux lines (Meissner–Ochsenfeld effect). This Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type-I superconductors, superconductivity is abruptly destroyed via a first order phase transition when the strength of the applied field rises above a critical value H . As such, they have only a c single critical magnetic field at which the material ceases to superconduct, becoming resistive. Depending on the geometry of the sample, one may obtain an intermediate state described first by [1] consisting of a baroque pattern[2] of macroscopically large regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. Elementary superconductors, such as aluminium and lead are typical Type-I superconductors. The origin of their superconductivity is explained by BCS theory. This type of superconductivity is normally exhibited by pure metals, e.g. aluminium, lead or mercury.

References

[1] L.D. Landau (1984). Electrodynamics of Continuous Media. Course of Theoretical Physics. Vol. 8. Butterworth-Heinemann. ISBN 0-7506-2634-8. [2] D.J.E. Callaway (1990). "On the remarkable structure of the superconducting intermediate state". Nuclear Physics B 344: 627–645. Bibcode 1990NuPhB.344..627C. doi:10.1016/0550-3213(90)90672-Z. Yttrium Wikibarium Book copper oxide Mounir Gmati 15 Yttrium barium copper oxide

Yttrium barium copper oxide

Identifiers

[1] CAS number 107539-20-8

Properties

Molecular formula YBa Cu O 2 3 7 Molar mass 666.19

Appearance Black solid

3[2] [3] Density 6.3 g/cm

Melting point >1000 °C

Solubility in water Insoluble

Structure

Crystal structure Based on the perovskite structure.

Coordination Orthorhombic geometry

Hazards

EU classification Irritant (Xi)

Related compounds

Related high-T BaLaO c 3-x superconductors

Related compounds Yttrium(III) oxide Barium oxide Copper(II) oxide

[4] (verify) (what is: / ?) Except where noted otherwise, data are given for materials in their standard state (at 25 °C, 100 kPa)

Infobox references

Yttrium barium copper oxide, often abbreviated YBCO, is a crystalline chemical compound with the formula YBa Cu O . This material, a famous "high-temperature superconductor", achieved prominence because it was the 2 3 7 first material to achieve superconductivity above the boiling point (77 K) of liquid nitrogen.

History In April 1986 (seventy-five years after the discovery of superconductivity in 1911), Georg Bednorz and Karl Müller, working at IBM in Zurich, discovered that certain semiconducting oxides became superconducting at 35 K, then considered a relatively high temperature. In particular, the lanthanum barium copper oxides, an oxygen deficient perovskite-related material, proved promising. In 1987, Bednorz and Müller were jointly awarded the Nobel Prize in Physics for this work. Yttrium Wikibarium Book copper oxide Mounir Gmati 16

Building on that, Maw-Kuen Wu and his graduate students, Ashburn and Torng[5] at the University of Alabama in Huntsville in 1987, and Paul Chu and his students at the University of Houston in 1987 (see superconductor page for info), discovered YBCO has a T of 93 K. (The first samples were Y Ba CuO .) Their work led to a rapid c 1.2 0.8 4 succession of new high temperature superconducting materials, ushering in a new era in material science and chemistry. YBCO was the first material to become superconducting above 77 K, the boiling point of liquid nitrogen. All materials developed before 1986 became superconducting only at temperatures near the boiling points of liquid helium (T = 4.2 K) or liquid hydrogen (T = 20.28 K) — the highest being Nb Ge at 23 K. The significance of the b b 3 discovery of YBCO is the much lower cost of the refrigerant used to cool the material to below the critical temperature.

Synthesis Relatively pure YBCO was first synthesized by heating a mixture of the metal carbonates at temperatures between 1000 to 1300 K.[6] [7] 4 BaCO + Y (CO ) + 6 CuCO + (1/2−x) O → 2 YBa Cu O + 13 CO 3 2 3 3 3 2 2 3 7−x 2 Modern syntheses of YBCO use the corresponding oxides and nitrates.[7] The superconducting properties of YBa Cu O are sensitive to the value of x, its oxygen content. Only those 2 3 7−x materials with 0 ≤ x ≤ 0.65 are superconducting below T , and when x ~ 0.07 the material superconducts at the c highest temperature of 95 K,[7] or in highest magnetic fields: 120 T for B perpendicular and 250 T for B parallel to the CuO planes.[8] 2 In addition to being sensitive to the stoichiometry of oxygen, the properties of YBCO are influenced by the crystallization methods used. Care must be taken to sinter YBCO. YBCO is a crystalline material, and the best superconductive properties are obtained when crystal grain boundaries are aligned by careful control of annealing and quenching temperature rates. Numerous other methods to synthesize YBCO have developed since its discovery by Wu and his coworkers, such as chemical vapor deposition (CVD),[6] [7] sol-gel,[9] and aerosol[10] methods. These alternative methods, however, still require careful sintering to produce a quality product. However, new possibilities have been opened since the discovery that trifluoroacetic acid (TFA), a source of fluorine, prevents the formation of the undesired barium carbonate (BaCO ). Routes such as CSD (chemical solution 3 deposition) have opened a wide range of possibilities, particularly in the preparation of long length YBCO tapes.[11] This route lowers the temperature necessary to get the correct phase to around 700 °C. This, and the lack of dependence on vacuum, makes this method a very promising way to get scalable YBCO tapes. Yttrium Wikibarium Book copper oxide Mounir Gmati 17

Structure

YBCO crystallises in a defect perovskite structure consisting of layers. The boundary of each layer is defined by planes of square planar CuO units 4 sharing 4 vertices. The planes can some times be slightly puckered.[6] Perpendicular to these CuO 2 planes are CuO ribbons sharing 2 vertices. The 4 yttrium atoms are found between the CuO planes, 2 while the barium atoms are found between the CuO 4 ribbons and the CuO planes. This structural feature 2 is illustrated in the figure to the right.

More details

Although YBa Cu O is a well-defined chemical 2 3 7 compound with a specific structure and stoichiometry, materials with less than seven oxygen atoms per formula unit are non-stoichiometric compounds. The structure of these materials depends on the oxygen content. This non-stoichiometry is denoted by the YBa Cu O in the chemical 2 3 7-x formula. When x = 1, the O(1) sites in the Cu(1) layer are vacant and the structure is tetragonal. The tetragonal form of YBCO is insulating and does not superconduct. Increasing the oxygen content slightly causes more of the O(1) sites to become occupied. For x < 0.65, Cu-O chains along the b axis of the crystal are formed. Elongation of the b axis changes the structure to orthorhombic, with lattice parameters of a = 3.82, b = 3.89, and c = 11.68 Å. Optimum superconducting properties occur when x ~ 0.07, i.e., almost all of the O(1) sites are occupied, with few vacancies.

In experiments where other elements are substituted at the Cu and Ba sites evidence has shown that conduction occurs in the Cu(2)O planes while the Cu(1)O(1) chains act as charge reservoirs, which provide carriers to the CuO planes. However, this model fails to address superconductivity in the homologue Pr123 (praseodymium instead of yttrium).[12] This (conduction in the copper planes) confines conductivity to the a-b planes and a large anisotropy in transport properties is observed. Along the c axis, normal conductivity is 10 times smaller than in the a-b plane. For other cuprates in the same general class, the anisotropy is even greater and inter-plane transport is highly restricted. Furthermore, the superconducting length scales show similar anisotropy, in both penetration depth (λ ≈ 150 nm, λ ab c ≈ 800 nm) and coherence length, (ξ ≈ 2 nm, ξ ≈ 0.4 nm). Although the coherence length in the a-b plane is 5 ab c times greater than that along the c axis it is quite small compared to classic superconductors such as niobium (where ξ ≈ 40 nm). This modest coherence length means that the superconducting state is more susceptible to local disruptions from interfaces or defects on the order of a single unit cell, such as the boundary between twinned crystal domains. This sensitivity to small defects complicates fabricating devices with YBCO, and the material is also sensitive to degradation from humidity. Yttrium Wikibarium Book copper oxide Mounir Gmati 18

Superconductive properties It is a Type-II superconductor. Penetration depth : 120 nm in the ab plane, 800 nm along the c axis. Coherence length : 2 nm in the ab plane, 0.4 nm along the c axis.

Properties of single crystals The upper critical field is 120 T for B perpendicular and 250 T for B parallel to the CuO planes. 2

Bulk properties Bulk properties depend greatly on the manner of synthesis and treatment because of the effect on crystal size, alignment, and density and type of lattice defects.

Applications in technology "The implementation of thin-film YBCO receiver coils has improved the signal-to-noise ratio of nuclear magnetic resonance (NMR) spectrometers by a factor of 3 compared to that achievable with conventional coils."[13] Several commercial applications of high temperature superconducting materials have been realized. For example, superconducting materials are finding use as magnets in magnetic resonance imaging, magnetic levitation, and Josephson junctions. (The most used material for power cables and magnets is BSCCO.) YBCO has yet to be used in many applications involving superconductors for two primary reasons: • First, while single crystals of YBCO have a very high critical current density, polycrystals have a very low critical current density: only a small current can be passed while maintaining superconductivity. This problem is due to crystal grain boundaries in the material. When the grain boundary angle is greater than about 5°, the supercurrent cannot cross the boundary. The grain boundary problem can be controlled to some extent by preparing thin films via CVD or by texturing the material to align the grain boundaries. • A second problem limiting the use of this material in technological applications is associated with processing of the material. Oxide materials such as this are brittle, and forming them into wires by any conventional process does not produce a useful superconductor. (Unlike BSCCO, the powder-in-tube process does not give good results with YBCO.) It should be noted that cooling materials to liquid nitrogen temperature (77 K) is often not practical on a large scale, although many commercial magnets are routinely cooled to liquid helium temperatures (4.2 K). The most promising method developed to utilize this material involves deposition of YBCO on flexible metal tapes coated with buffering metal oxides. This is known as coated conductor. Texture (crystal plane alignment) can be introduced into the metal tape itself (the RABiTS process) or a textured ceramic buffer layer can be deposited, with the aid of an ion beam, on an untextured alloy substrate (the IBAD process). Subsequent oxide layers prevent diffusion of the metal from the tape into the superconductor while transferring the template for texturing the superconducting layer. Novel variants on CVD, PVD, and solution deposition techniques are used to produce long lengths of the final YBCO layer at high rates. Companies pursuing these processes include American Superconductor, Superpower (a division of Intermagnetics General Corp), Sumitomo, Fujikura, Nexans Superconductors, and European Advanced Superconductors. A much larger number of research institutes have also produced YBCO tape by these methods. Yttrium Wikibarium Book copper oxide Mounir Gmati 19

Surface modification of YBCO Surface modification of materials has often led to new and improved properties. Corrosion inhibition, adhesion and nucleation, preparation of organic superconductor/insulator/high-T superconductor trilayer structures, c and the fabrication of metal/insulator/superconductor tunnel junctions have been developed using surface-modified YBCO.[14] These molecular layered materials are synthesized using cyclic voltammetry. Thus far, YBCO layered with alkylamines, arylamines, and thiols have been produced with varying stability of the molecular layer. It has been proposed that amines act as Lewis bases and bind to Lewis acidic Cu surface sites in YBa Cu O to form stable 2 3 7 coordination bonds.

References

[1] http:/ / www. commonchemistry. org/ ChemicalDetail. aspx?ref=107539-20-8 [2] Knizhnik, A (2003). "Interrelation of preparation conditions, morphology, chemical reactivity and homogeneity of ceramic YBCO". Physica C: Superconductivity 400: 25. Bibcode 2003PhyC..400...25K. doi:10.1016/S0921-4534(03)01311-X. [3] Grekhov, I (1999). "Growth mode study of ultrathin HTSC YBCO films on YBaCuNbO buffer". Physica C: Superconductivity 324: 39. Bibcode 1999PhyC..324...39G. doi:10.1016/S0921-4534(99)00423-2.

[4] http:/ / en. wikipedia. org/ wiki/ %3Ayttrium_barium_copper_oxide?diff=cur& oldid=432929974 [5] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu (1987). "Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O Compound System at Ambient Pressure". Physical Review Letters 58 (9): 908–910. Bibcode 1987PhRvL..58..908W. doi:10.1103/PhysRevLett.58.908. PMID 10035069. [6] Housecroft, C. E.; Sharpe, A. G. (2004). Inorganic Chemistry (2nd ed.). Prentice Hall. ISBN 978-0130399137. [7] Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Oxford: Butterworth-Heinemann. ISBN 0080379419.

[8] T. Sekitania, N. Miura, S. Ikedaa, Y. H. Matsudaa, Y. Shioharab (2004). "Upper critical field for optimally-doped YBa2Cu3O7−δ" (http:/ /

www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B6TVH-4BWYV0D-H& _user=10& _rdoc=1& _fmt=& _orig=search&

_sort=d& view=c& _acct=C000050221& _version=1& _urlVersion=0& _userid=10& md5=28aeb1ca959e86bc4a201f483224ec06). Elsevier Science B.V.. . [9] Yang-Kook Sun, In-Hwan Oh (1996). "Preparation of Ultrafine YBa2Cu3O7-x Superconductor Powders by the Poly(vinyl alcohol)-Assisted Sol−Gel Method". Ind. Eng. Chem. Res. 35 (11): 4296. doi:10.1021/ie950527y. [10] Zhou, Derong (1991) (Ph.D. Thesis). Yttrium Barium Copper Oxide Superconducting Powder Generation by An Aerosol Process. University of Cincinnati. pp. 28. Bibcode 1991PhDT...... 28Z. [11] O. Castano, A. Cavallaro, A. Palau, J. C. Gonzalez, M. Rossell, T. Puig, F. Sandiumenge, N. Mestres, S. Pinol, A. Pomar, and X. Obradors (2003). "High quality YBa Cu O thin films grown by trifluoroacetates metal-organic deposition". Supercond. Sci. Technol. 16: 45–53. 2 3 {7–x} Bibcode 2003SuScT..16...45C. doi:10.1088/0953-2048/16/1/309. [12] Oka, K (1998). "Crystal growth of superconductive PrBa2Cu3O7−y". Physica C 300 (3–4): 200. doi:10.1016/S0921-4534(98)00130-0.

[13] Superconducting devices (http:/ / encyclopedia2. thefreedictionary. com/ Superconducting+ devices) [14] F. Xu et al. (1998). "Surface Coordination Chemistry of YBa2Cu3O7-δ". Langmuir 14 (22): 6505. doi:10.1021/la980143n.

External links

• Diagram of YBCO structure (http:/ / www. ch. ic. ac. uk/ otway/ YBCO. html)

• New World Record For Superconducting Magnet 26.8T April 2007 (http:/ / www. physorg. com/

news105718161. html)

• External MSDS Data Sheet (safety classifications) for YBCO (http:/ / www. futurescience. com/ manual/

ybcomsds. html).

• Superconductivity in everyday life : Interactive exhibition – little if any specific to YBCO (http:/ / www.

superlife. info). Liquid nitrogen Wiki Book Mounir Gmati 20 Liquid nitrogen

Liquid nitrogen is nitrogen in a liquid state at a very low temperature. It is produced industrially by fractional distillation of liquid air. Liquid nitrogen is a colourless clear liquid with density of 0.807 g/mL at its boiling point and a dielectric constant of 1.4.[1] Liquid nitrogen is often referred to by the abbreviation, LN or "LIN" or "LN" and has the UN 2 number 1977.

At atmospheric pressure, liquid nitrogen boils at 77K (-196°C; -321°F) and is a cryogenic fluid which can cause rapid freezing on contact with living tissue, which may lead to frostbite. When appropriately insulated from ambient heat, liquid nitrogen can be stored and transported, for example in vacuum flasks. Here, the very low temperature is held constant at 77 K by slow boiling of the liquid, resulting in the Liquid nitrogen evolution of nitrogen . Depending on the size and design, the holding time of vacuum flasks ranges from a few hours to a few weeks.

Liquid nitrogen can easily be converted to the solid by placing it in a vacuum chamber pumped by a rotary vacuum pump.[2] Liquid nitrogen freezes at 63 K (−210 °C; −346 °F). Despite its reputation, liquid nitrogen's efficiency as a coolant is limited by the fact that it boils immediately on contact with a warmer object, enveloping the object in insulating nitrogen gas. This effect, known as the Leidenfrost effect, applies to any liquid in contact with an object significantly hotter than its boiling point. More rapid cooling may be obtained by plunging an object into a slush of liquid and solid nitrogen than into liquid nitrogen alone. MIT students preparing homemade ice cream with liquid nitrogen. Nitrogen was first liquefied at the Jagiellonian University on 15 April 1883 by Polish physicists, Zygmunt Wróblewski and Karol Olszewski.[3]

Uses Liquid nitrogen is a compact and readily transported source of nitrogen gas without pressurization. Further, its ability to maintain temperatures far below the freezing point of water makes it extremely useful in a wide range of applications, primarily as an open-cycle refrigerant, including: • in cryotherapy for removing unsightly or potentially malignant skin lesions such as warts and actinic keratosis • as a coolant for CCD cameras in astronomy • to store cells at low temperature for laboratory work • in cryogenics • as a source of very dry nitrogen gas • for the immersion freezing and transportation of food products • for the cryopreservation of blood, reproductive cells (sperm and egg), and other biological samples and materials • as a method of freezing water pipes in order to work on them in situations where a valve is not available to block water flow to the work area • in the process of promession, a way to dispose of the dead • for cooling a high-temperature superconductor to a temperature sufficient to achieve superconductivity Liquid nitrogen Wiki Book Mounir Gmati 21

• for the cryonic preservation in the hope of future reanimation. • to preserve tissue samples from surgical excisions for future studies • to shrink-weld machinery parts together • as a coolant for vacuum pump traps and in controlled-evaporation processes in chemistry. • as a coolant to increase the sensitivity of infrared homing seeker heads of missiles such as the Strela 3 • as a coolant to temporarily shrink mechanical components during machine assembly and allow improved interference fits • as a coolant for computers[4] • in food preparation, such as for making ultra-smooth ice cream.[5]

Safety

Since the liquid to gas expansion ratio of nitrogen is 1:694 at 20C, a tremendous amount of force can be generated if liquid nitrogen is rapidly vaporized. In an incident in 2006 at Texas A&M University, the pressure-relief devices of a tank of liquid nitrogen were malfunctioning and later sealed. As a result of the subsequent pressure buildup, the tank failed catastrophically and exploded. The force of the explosion was sufficient to propel the tank through the ceiling immediately above it.[6]

Because of its extremely low temperature, careless handling of liquid nitrogen may result in cold burns. As liquid nitrogen evaporates it will reduce the oxygen concentration in the air and might act as an asphyxiant, especially in confined spaces. Nitrogen is odorless, colorless and tasteless, and may produce asphyxia without any sensation or prior warning.[7] A laboratory assistant died in Scotland in 1999, apparently from asphyxiation, possibly caused by liquid nitrogen spilled in a basement storage room.[8] Filling a liquid nitrogen Dewar from a storage Vessels containing liquid nitrogen can condense oxygen from air. The tank liquid in such a vessel becomes increasingly enriched in oxygen (boiling point = 90 K) as the nitrogen evaporates, and can cause violent oxidation of organic material.

References

[1] "Dielectric Constants" (http:/ / www. apgsensors. com/ ltr2/ access. php?file=pdf/ dielectric-constants. pdf). . [2] Umrath, W. (1974) Cooling bath for rapid freezing in electron microscopy. Journal of Microscopy 101, 103–105.

[3] William Augustus Tilden (2009). A Short History of the Progress of Scientific Chemistry in Our Own Times (http:/ / books. google. com/

books?id=8SKrWdFLEd4C& pg=PA249). BiblioBazaar, LLC. p. 249. ISBN 1103358421. . [4] Wainner, Scott; Robert Richmond (2003). The Book of Overclocking: Tweak Your PC to Unleash Its Power. No Starch Press. pp. 44. ISBN 188641176X.

[5] Liquid Nitrogen Ice Cream Recipe (http:/ / www. 101cookbooks. com/ archives/ 001366. html), March 7, 2006

[6] Brent S. Mattox. "Investigative Report on Chemistry 301A Cylinder Explosion" (http:/ / ucih. ucdavis. edu/ docs/ chemistry_301a. pdf) (reprint). Texas A&M University. .

[7] British Compressed Association (2000) BCGA Code of Practice CP30. The Safe Use of Liquid nitrogen Dewars up to 50 litres. (http:/ /

www. bcga. co. uk/ preview/ products. php?g1=3ff921& n=2) ISSN 0260-4809.

[8] Inquiry after man dies in chemical leak (http:/ / news. bbc. co. uk/ 2/ hi/ uk_news/ scotland/ 484813. stm), BBC News, . Liquid nitrogen Wiki Book Mounir Gmati 22

External links

• Behind the scenes video (http:/ / www. younghotelier. com/ interviews/

video-cooking-with-liquid-nitrogen-in-the-real-world-tang-restaurant-dubai/ ) – How liquid nitrogen is used in restaurants for cooking and cocktails Wiki Book Mounir Gmati 23

Magnetism

Magnetism

Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well as the materials that are attracted to them. However, all materials are influenced to a greater or lesser degree by the presence of a magnetic field. Some are attracted to a magnetic field (); others are repulsed by a magnetic field (diamagnetism); others have a much more complex relationship with an applied magnetic field ( behavior and ). Substances that are negligibly affected by magnetic fields are known as non-magnetic substances. They include copper, aluminium, gases, and plastic. The magnetic state (or phase) of a material depends on temperature (and other variables such as pressure and applied magnetic field) so that a material may exhibit more than one form of magnetism depending on its temperature, etc.

History Aristotle attributed the first of what could be called a scientific discussion on magnetism to Thales, who lived from about 625 BCE to about 545 BCE.[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 BCE book called Book of the Devil Valley Master (鬼谷子): "The lodestone makes iron come or it attracts it."[3] The earliest mention of the attraction of a needle appears in a work composed between AD 20 and 100 (Louen-heng): "A lodestone attracts a needle."[4] The ancient Chinese scientist Shen Kuo (1031–1095) was the first person to write of the magnetic needle compass and that it improved the accuracy of navigation by employing the astronomical concept of true north (Dream Pool Essays, AD 1088), and by the 12th century the Chinese were known to use the lodestone compass for navigation. They sculpted a directional spoon from lodestone in such a way that the handle of the spoon always 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 of magnets. In 1282, the properties of magnets and the dry compass were discussed by Al-Ashraf, a Yemeni physicist, astronomer, and geographer.[5] In 1600, William Gilbert published his De 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 the terrella. From his experiments, he concluded that the Earth was itself magnetic and that this was the reason compasses pointed north (previously, some believed that it was the pole star (Polaris) or a large magnetic island on the north pole that attracted the compass). An understanding of the relationship between electricity and magnetism began in 1819 with work by Hans Christian Oersted, a professor at the University of Copenhagen, who discovered more or less by accident that an electric current could influence a compass needle. This landmark experiment is known as Oersted's Experiment. Several other experiments followed, with André-Marie Ampère, who in 1820 discovered that the magnetic field circulating in a closed-path was related to the current flowing through the perimeter of the path; Carl Friedrich Gauss; Jean-Baptiste Biot and Félix Savart, both of which in 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, and others finding further links between magnetism and electricity. Magnetism Wiki Book Mounir Gmati 24

James Clerk Maxwell synthesized and expanded these insights into Maxwell's equations, unifying electricity, magnetism, and optics into the field of . In 1905, Einstein used these laws in motivating his theory of special relativity,[6] requiring that the laws held true in all inertial reference frames. Electromagnetism has continued to develop into the 21st century, being incorporated into the more fundamental theories of gauge theory, quantum electrodynamics, electroweak theory, and finally the standard model.

Sources of magnetism Magnetism, 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. In magnetic materials, sources of magnetization are the electrons' orbital angular motion around the nucleus, and the electrons' intrinsic magnetic moment (see electron magnetic dipole moment). The other sources of magnetism are the nuclear magnetic moments of the nuclei in the material which 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 important in other contexts, particularly in nuclear magnetic resonance (NMR) and magnetic 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 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 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 electron magnetic moments will be, on average, lined up. Then the material can produce a net total magnetic field, which can potentially be quite strong. The magnetic behavior of a material depends on its structure, particularly its electron configuration, for the reasons mentioned above, and also on the temperature. At high temperatures, random thermal motion makes it more difficult for the electrons to maintain alignment.

Topics Magnetism Wiki Book Mounir Gmati 25

Diamagnetism

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.[8] Thus, despite its [7] universal occurrence, diamagnetic Hierarchy of types of magnetism. 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 understood classically as follows:

When a material is put in a magnetic field, the electrons circling the nucleus will experience, in addition to their Coulomb attraction to the nucleus, a Lorentz force from the magnetic field. Depending on which direction the electron is orbiting, this force may increase the centripetal 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 with Lenz's law). This results in a small bulk magnetic moment, with an opposite direction to the applied field. Note that this description is meant only as an heuristic; a proper understanding requires a quantum-mechanical description. Note that 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.

Paramagnetism In a paramagnetic material there are unpaired electrons, i.e. atomic or molecular orbitals with exactly one electron in them. While paired electrons are required by the Pauli 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.

Ferromagnetism 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 when the applied field is removed, the electrons in the material maintain a parallel orientation. Every ferromagnetic substance has its own individual temperature, called the Curie 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. Some well-known ferromagnetic materials that exhibit easily detectable magnetic properties (to form magnets) are nickel, iron, cobalt, gadolinium and their alloys. Magnetism Wiki Book Mounir Gmati 26

Magnetic domains

The magnetic moment of atoms in a ferromagnetic 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 called magnetic domains or Weiss domains. Magnetic domains can be observed with a magnetic 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.

Magnetic domains in ferromagnetic material.

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 as shown at the right. When exposed to a magnetic field, the domain boundaries move so that the domains aligned with the magnetic field grow and dominate the structure 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 is magnetically saturated. When a magnetized ferromagnetic material is heated to the Curie point temperature, the molecules are Effect of a magnet on the domains. 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 can freeze into a crystalline solid.

Antiferromagnetism

In an antiferromagnet, unlike a ferromagnet, there is a tendency for the intrinsic magnetic moments of neighboring valence electrons to point in opposite directions. When all atoms are arranged in a substance so that each neighbor is 'anti-aligned', the substance is antiferromagnetic. Antiferromagnets have a zero net magnetic moment, meaning no field is produced by them. Antiferromagnets are less common compared to the other types of behaviors, and are mostly Antiferromagnetic ordering observed 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 geometrical arrangement in which each pair of neighbors is anti-aligned. This is called a spin glass, and is an example of geometrical frustration. Magnetism Wiki Book Mounir Gmati 27

Ferrimagnetism

Like ferromagnetism, ferrimagnets retain their magnetization in the absence of a field. However, like antiferromagnets, neighboring pairs of electron spins like 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. Ferrimagnetic ordering

The first discovered magnetic substance, magnetite, was originally believed to be a ferromagnet; Louis Néel disproved this, however, with the discovery of .

Superparamagnetism When a ferromagnet or ferrimagnet is sufficiently small, it acts like a single magnetic spin that is subject to Brownian motion. Its response to a magnetic field is qualitatively similar to the response of a paramagnet, but much larger.

Electromagnet An electromagnet is a type of magnet whose magnetism is produced by the flow of electric current. The magnetic field disappears when the current ceases.

Other types of magnetism

• Molecular magnet • Metamagnetism • Molecule-based magnet • Spin glass

Magnetism, electricity, and special relativity

Electromagnets attracts paper clips when current As a consequence of Einstein's theory of special relativity, electricity is applied creating a magnetic field. The and magnetism are fundamentally interlinked. Both magnetism lacking electromagnet loses them when current and electricity, and electricity without magnetism, are inconsistent with magnetic field are removed. special relativity, due to such effects as length contraction, time dilation, and the fact that the magnetic force is velocity-dependent. However, when both electricity and magnetism are taken into account, the resulting theory (electromagnetism) is fully consistent with special relativity.[6] [9] In particular, a phenomenon that appears purely electric to one observer may be purely magnetic 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 called electromagnetism, analogous to how relativity "mixes" space and time into spacetime. Magnetism Wiki Book Mounir Gmati 28

Magnetic fields in a material In a vacuum,

where μ is the vacuum permeability. 0 In a material,

The quantity μ M is called magnetic polarization. 0 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 is zero (see Remanence).

Force due to magnetic field - The magnetic force

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 the Biot-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 electrically charged particles are in motion---for example, from movement of Magnetic lines of force of a bar magnet shown by electrons in an electric current, or in certain cases from the orbital iron filings on paper motion 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 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. When a charged particle moves through a magnetic field B, it feels a Lorentz force F given by the cross product:[10]

where is the electric charge of the particle, and v is the velocity vector of the particle Because this is a cross product, the force is perpendicular to both the motion of the particle and the magnetic field. It follows 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 force exerted is labeling the index finger "V", the middle finger "B", and the thumb "F" with your right hand. When making a gun-like configuration, with the middle finger crossing under the index finger, the fingers represent the Magnetism Wiki Book Mounir Gmati 29

velocity vector, magnetic field vector, and force vector, respectively. See also right hand rule.

Magnetic dipoles A very common source of magnetic field shown in nature is a dipole, with a "South pole" and a "North pole", terms dating back to the use of magnets as compasses, interacting with the Earth's magnetic field to indicate North and South on the globe. Since opposite ends of magnets are attracted, the north pole of a magnet is attracted to the south pole of another magnet. The Earth's North 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 contains energy, and physical systems move toward configurations with lower energy. When diamagnetic material is placed in a magnetic field, a magnetic 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.

Magnetic monopoles 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, 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 Dirac observed in 1931 that, because electricity and magnetism show a certain symmetry, just as quantum theory predicts that individual positive or negative 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 observed elementary particles carry charges 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 in the big 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 of inflation (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.[12]

Quantum-mechanical origin of magnetism In principle all kinds of magnetism originate (similar to Superconductivity) from specific quantum-mechanical phenomena which are not easily explained (e.g. Mathematical formulation of quantum mechanics, in particular the chapters on spin and on the Pauli principle). A successful model was developed already in 1927, by Walter Heitler and Fritz London, who derived quantum-mechanically, how hydrogen molecules are formed from hydrogen atoms, i.e. from the atomic hydrogen orbitals 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 resulting orbital is: Magnetism Wiki Book Mounir Gmati 30

Here the last product means that a first electron, r , is in an atomic hydrogen-orbital centered at the second nucleus, 1 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 of chemical bonds, but as we will see, also for magnetism, i.e. in this connection the term exchange 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 the spin 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), 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 r by the discrete values s (= ±½); thereby we have i i and . The "singlet state", i.e. the - sign, means: the spins are antiparallel, i.e. for the solid we have antiferromagnetism, and for two-atomic molecules one has diamagnetism. The tendency to form a (homoeopolar) chemical bond (this means: the formation of a symmetric molecular 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 each other 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 are ferromagnetic. 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 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 the Heisenberg 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.

Units of electromagnetism

SI units related to magnetism

SI electromagnetism units

[13] Symbol 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 Magnetism Wiki Book Mounir Gmati 31

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

Other units • gauss – The gauss, abbreviated as G, is the CGS unit of magnetic field (B). • oersted – The oersted is the CGS unit of magnetizing field (H). • Maxwell – is the CGS unit for the magnetic flux. • gamma – is a unit of magnetic flux density that was commonly used before the tesla became popular (1 gamma = 1 nT) • μ – common symbol for the permeability of free space (4π×10−7 N/(ampere-turn)2). 0

Living things Some organisms can detect magnetic fields, a phenomenon known as magnetoception. Magnetobiology studies magnetic 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 0748406603. .

[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 0632042052. . [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:/ / www. iupac. org/ publications/ books/ gbook/

green_book_2ed. pdf) Magnetism Wiki Book Mounir Gmati 32

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.

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))

• Magnetism Experiments (http:/ / sciencecastle. com/ sc/ index. php/ scienceexperiments/ search?p=0& t=a&

v=mr& c=0& cl=1)

• 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. Magnetic Wiki field Book Mounir Gmati 33 Magnetic field

A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field.[1] The magnetic field is most commonly defined in terms of the Lorentz force it exerts on moving electric charges. There are two separate but closely related fields to which the name 'magnetic field' can refer: a magnetic B field and a magnetic H field. Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In special relativity, electric and magnetic fields are two interrelated aspects of a single object, called the electromagnetic field tensor; the aspect of the electromagnetic field that is seen as a magnetic field is dependent on the reference frame of the observer. In quantum physics, the electromagnetic field is quantized and electromagnetic interactions result from the exchange of photons. 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 . The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits.

History

Although magnets and magnetism were known much earlier, one of the first descriptions of the magnetic field was produced in 1269 by the French scholar Petrus Peregrinus[2] who mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting that the resulting field lines crossed at two points he named those points 'poles' in analogy to Earth's poles. Almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus' work and was the first to state explicitly that Earth itself was a magnet. Gilbert's great work De Magnete was published in 1600 and helped to establish the study of magnetism as a science. One of the first drawings of a magnetic field, by René Descartes, 1644. It illustrated his theory that magnetism was caused by the circulation of tiny helical One of the first successful models of the particles, "threaded parts", through threaded pores in magnets. magnetic field was developed in 1824 by Siméon-Denis Poisson (1781–1840). Poisson assumed that magnetism was due to 'magnetic charges' such that like 'magnetic charges' repulse while opposites attract. The model he created is exactly analogous to modern electrostatics with a magnetic H-field being produced by 'magnetic charges' in the same way that an electric field E-field is produced by electric charges. It predicts the correct H-field for permanent magnets. It predicts the forces between magnets. And, it predicts the correct energy stored in the magnetic fields.[3]

Three remarkable discoveries though, would challenge Poisson's model. First, in 1819, Hans Christian Oersted discovered that an electric current generates a magnetic field encircling it. Then, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally Jean-Baptiste Biot and Félix Savart Magnetic Wiki field Book Mounir Gmati 34

discovered the Biot–Savart law which correctly predicts the magnetic field around any current-carrying wire. Together, these discoveries suggested a model in which the magnetic B field of a material is produced by microscopic current loops. In this model, these current loops (called magnetic dipoles) would replace the dipoles of charge of the Poisson's model.[4] No magnetic charges are needed which has the additional benefit of explaining why magnetic charge can not be isolated; cutting a magnet in half does not result in two separate poles but in two separate magnets, each of which has both poles. The next decade saw two developments that help lay the foundation for the full theory of electromagnetism. In 1825, Ampère published his Ampère's law which like the Biot–Savart law correctly described the magnetic field generated by a steady current but was more general. And, in 1831, Michael Faraday showed that a changing magnetic field generates an encircling electric field and thereby demonstrated that electricity and magnetism are even more tightly knitted. Between 1861 and 1865, James Clerk Maxwell developed and published a set of Maxwell's equations which explained and united all of classical electricity and magnetism. The first set of these equations was published in a paper entitled On Physical Lines of Force in 1861. The mechanism that Maxwell proposed to underlie these equations in this paper was fundamentally incorrect, which is not surprising since it predated the modern understanding even of the atom. Yet, the equations were valid although incomplete. He completed the set of Maxwell's equations in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrated the fact that light is an electromagnetic wave. Thus, he theoretically unified not only electricity and magnetism but light as well. This fact was then later confirmed experimentally by Heinrich Hertz in 1887. Even though the classical theory of electrodynamics was essentially complete with Maxwell's equations, the twentieth century saw a number of improvements and extensions to the theory. Albert Einstein, in his great paper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the same phenomena viewed from different reference frames. (See moving magnet and conductor problem for details about the thought experiment that eventually helped Albert Einstein to develop special relativity.) Finally, the emergent field of quantum mechanics was merged with electrodynamics to form quantum electrodynamics or QED.

Definitions, units, and measurement

[5] Alternative names for the field B

• Magnetic flux density • Magnetic induction • Magnetic field

[5] [6] Alternative names for the field H

• Magnetic field intensity • Magnetic field strength • Magnetic field • Magnetizing field

The term magnetic field is historically used to describe a magnetic H-field whereas other terms were used to describe a related magnetic B-field. Informally, and formally for some recent textbooks mostly in physics, the term 'magnetic field' is used to describe the magnetic B-field as well as or in place of H.[7] There are many alternative names for both (see sidebar to right). To avoid confusion, this article uses B-field and H-field for these fields, and uses magnetic field where either or both fields apply. The magnetic field can be defined in many equivalent ways based on the effects it has on its environment. For instance, a particle having an electric charge, q, and moving in a B-field with a velocity, v, experiences a force, F, called the Lorentz force. See Force on a charged particle below. Alternatively, the magnetic field can be defined in Magnetic Wiki field Book Mounir Gmati 35

terms of the torque it produces on a magnetic dipole. See Torque on a magnet due to a B-field below. The H-field is defined as a modification of B due to magnetic fields produced by material media. See H and B inside and outside of magnetic materials below for the relationship between B and H. Outside of a material (i.e., in vacuum) the B and H fields are indistinguishable. (They only differ by a multiplicative constant.) Inside a material, though, they may differ in relative magnitude and even direction. Often, though, they differ only by a material dependent multiplicative constant. The B-field is measured in teslas in SI units and in gauss in cgs units. (1 tesla = 10,000 gauss). The SI unit of tesla is equivalent to (newton·second)/(coulomb·metre).[8] The H-field is measured in ampere-turn per metre (A/m) in SI units, and in oersteds (Oe) in cgs units.[9] Devices used to measure the local magnetic field are called magnetometers. Important classes of magnetometers include using a rotating coil, Hall effect magnetometers, NMR magnetometers, SQUID magnetometers, and fluxgate magnetometers. The magnetic fields of distant astronomical objects are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce synchrotron radiation which is detectable in radio waves. The smallest precision level for a magnetic field measurement[10] is on the order of attoteslas (10−18 tesla); the largest magnetic field produced in a laboratory is 2,800 T (VNIIEF in Sarov, Russia, 1998).[11] The magnetic field of some astronomical objects such as magnetars are much higher; magnetars range from 0.1 to 100 GT (108 to 1011 T).[12] See orders of magnitude (magnetic field).

Magnetic field lines

Mapping the magnetic field of an object is simple in principle. First, measure the strength 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 the local magnetic field with a length proportional to the strength of the magnetic field. A simpler way to visualize the magnetic field is to 'connect' the arrows to form "magnetic field lines". Magnetic field lines make it much easier to visualize and understand the complex mathematical relationships underlying magnetic field. If done carefully, a field line diagram contains the same information as the vector field it represents. The magnetic field can be estimated at any point on a magnetic field line diagram (whether on a field line or not) using the direction and density of nearby magnetic field lines.[13] A higher density of nearby field lines indicates a stronger magnetic field.

Compasses reveal the direction of the local magnetic field. As seen here, the magnetic field points towards a magnet's south pole and away from its north pole. Magnetic Wiki field Book Mounir Gmati 36

Various phenomena have the effect of "displaying" magnetic field lines as though the field lines are physical phenomena. For example, iron filings placed in a magnetic field line up to form lines that correspond to 'field lines'. Magnetic fields "lines" are also visually displayed in polar auroras, in which particle dipole interactions create visible streaks of light that line up with the local direction of Earth's magnetic field. However, field lines are a visual and conceptual aid only and are no more real than (for example) the contour lines (constant altitude) on a topographic map. They do not exist in the The direction of magnetic field lines represented by the alignment of iron filings sprinkled on actual field; a different choice of mapping scale could show twice as paper placed above a bar magnet. The mutual many "lines" or half as many. attraction of opposite poles of the iron filings results in the formation of elongated clusters of Field lines can be used as a qualitative tool to visualize magnetic filings along "field lines". The field is not forces. In ferromagnetic substances like iron and in plasmas, magnetic precisely the same as around the isolated magnet; forces can be understood by imagining that the field lines exert a the magnetization of the filings alters the field tension, (like a rubber band) along their length, and a pressure somewhat. perpendicular to their length on neighboring field lines. 'Unlike' poles of magnets attract because they are linked by many field lines; 'like' poles repel because their field lines do not meet, but run parallel, pushing on each other.

Most physical phenomena that "display" magnetic field lines do not include which direction along the lines that the magnetic field is in. A compass, though, reveals that magnetic field lines outside of a magnet point from the north pole (compass points away from north pole) to the south (compass points toward the south pole). The magnetic field of a straight current-carrying wire encircles the wire with a direction that depends on the direction of the current and that can be measured with a compass as well.

B-field lines never end Field lines are a useful way to represent any vector field and often reveal sophisticated properties of fields quite simply. One important property of the B-field revealed this way is that magnetic B field lines neither start nor end (mathematically, B is a solenoidal vector field); a field line either extends to infinity or wraps around to form a closed curve.[14] To date no exception to this rule has been found. (See below.) Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet B-field lines continue through the magnet from the south pole back to the north.[15] If a B-field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point. Magnetic poles, therefore, always come in N and S pairs. Cutting a magnet in half results in two separate magnets each with both a north and a south pole. More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the 'number'[16] of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent to:

,

where the integral is a surface integral over the closed surface S (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since dA points outward, the dot product in the integral is positive for 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. Magnetic Wiki field Book Mounir Gmati 37

H-field lines begin and end near magnetic poles Unlike B-field lines, which never end, the H-field lines due to a magnetic material begin in a region(s) of the magnet called the north pole pass through the magnet and/or outside of the magnet and ends in a different region of the material called the south pole. Near the north pole, therefore, all H-field lines point away from the north pole (whether inside the magnet or out) while near the south pole (whether inside the magnet or out) all H-field lines point toward the south pole. (The B-field lines, for comparison, form a closed loop going from south to north inside the magnet and from north to south outside the magnet) The H-field, therefore, is analogous to the electric field E which starts at a positive charge and ends at a negative charge. It is tempting, therefore, to model magnets in terms of magnetic charges localized near the poles. Unfortunately, this model is incorrect; for instance, it often fails when determining the magnetic field inside of magnets. (See "Non-uniform magnetic field causes like poles to repel and opposites to attract" below.) Outside a material, though, the H-field is identical to the B-field (to a multiplicative constant) so that in many cases the distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials.

Magnetic monopole (hypothetical) A magnetic monopole is a hypothetical particle (or class of particles) that has, as its name suggests, only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end. Modern interest in this concept stems from particle theories, notably Grand Unified Theories and superstring theories, that predict either the existence, or the possibility, of magnetic monopoles. These theories and others have inspired extensive efforts to search for monopoles. Despite these efforts, no magnetic monopole has been observed to date.[17] In recent research, materials known as spin ices can simulate monopoles, but do not contain actual monopoles.

The magnetic field and permanent magnets Permanent magnets are objects that produce their own persistent magnetic fields. They are made of ferromagnetic materials, such as iron and nickel, that have been magnetized, and they have both a north and a south pole.

Magnetic field of permanent magnets The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The B field of a small[18] straight magnet is proportional to the magnet's strength (called its magnetic dipole moment m). The equations are non-trivial and also depend on the distance from the magnet and the orientation of the magnet. For simple magnets, m points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its m by 180 degrees. It is sometimes useful to model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as the Coulomb force between electric charges. In this model, a magnetic H-field is produced by magnetic charges that are 'smeared' around each pole. A north pole therefore feels a force in the direction of the H-field while the force on the south pole is opposite to the H-field. Unfortunately, the concept of poles of 'magnetic charge' does not accurately reflect what happens inside a magnet (see ferromagnetism). Magnetic charges do not exist. Magnetic poles cannot exist apart from each other; all magnets have north/south pairs which cannot be separated without creating two magnets each having a north/south pair. Finally, magnetic charge fails to account for magnetism that is produced by electric currents nor the force that a magnetic field applies to moving electric charges. Magnetic Wiki field Book Mounir Gmati 38

The more physically correct description of magnetism involves atomic sized loops of current distributed throughout the magnet.[19]

Non-uniform magnetic field causes like poles to repel and opposites to attract The force between two small magnets is quite complicated and depends on the strength and orientation of both magnets and the distance and direction of the magnets relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field B[20] of the other. To understand the force between magnets and to generalize it to other cases, it is useful to examine the magnetic charge model given above (with the caveats given above as well). In this model, the H-field of the first magnet pushes and pulls on the magnetic charges near both poles of the second magnet. If the H-field due to the first magnet is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. The magnetic field is not the same, though; the magnetic field is significantly stronger near the poles of a magnet. In this nonuniform magnetic field, each pole sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque. This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field whether caused by permanent magnets or by electric currents will exert a force on a small magnet in this way. Mathematically, the force on a small magnet having a magnetic moment m due to a magnetic field B is:[21]

where the gradient ∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. To understand this equation, note that the dot product m · B = mBcos(θ), where m and B represent the 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 dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions having their own m then summing up the forces on each of these regions.

Torque on a magnet due to a B-field Magnetic torque on a magnet due to an external magnetic field can be observed by placing two magnets near each other while allowing one to rotate. Magnetic torque is used to drive simple electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of electromagnets. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft. See Rotating magnetic fields below. Magnetic torque τ tends to align a magnet's poles with the B-field lines (since m is in the direction of the poles this is equivalent to saying that it tends to align m in the same direction as B). This is why the magnetic needle of a compass points toward earth's north pole. By definition, the direction of the Earth's local magnetic field is the direction in which the north pole of a compass (or of any magnet) tends to point. Mathematically, the torque τ on a small magnet is proportional both to the applied B-field and to the magnetic moment m of the magnet:

where × represents the vector cross product. Note that this equation includes all of the qualitative information included above. There is no torque on a magnet if m is in the same direction as B. (The cross product is zero for two Magnetic Wiki field Book Mounir Gmati 39

vectors that are in the same direction.) Further, all other orientations feel a torque that twists them toward the direction of B.

The magnetic field and electric currents Currents of electric charges both generate a magnetic field and feel a force due to magnetic B-fields.

Magnetic field due to moving charges and electric currents

All moving charged particles produce magnetic fields. Moving point charges, such as electrons, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.[22] Magnetic field lines form in concentric circles around a cylindrical current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the "right hand grip rule" (see figure at right). The strength of the magnetic field decreases

with distance from the wire. (For an infinite length wire the strength Right hand grip rule: current (I) flowing through decreases inversely proportional to the distance.) a conductor in the direction indicated by the white arrow produces a magnetic field (B) around the conductor as shown by the red arrows.

Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or "solenoid" enhances this effect. A device so formed around an iron core may act as an electromagnet, generating a strong, well-controlled magnetic field. An Solenoid infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil.

The magnetic field generated by a steady current (a constant flow of electric charges in which charge is neither accumulating nor depleting at any point)[23] is described by the Biot–Savart law:

where the integral sums over the wire length where vector dℓ is the direction of the current, μ is the magnetic 0 constant, 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[24] [25] way of relating the current to the B-field is through Ampère's law:

where the line integral is over any arbitrary loop and is the current enclosed by that loop. Ampère's law is enc always valid for steady currents and can be used to calculate the B-field for certain highly symmetric situations such as an infinite wire or an infinite solenoid. In a modified form that accounts for time varying electric fields, Ampère's law is one of four Maxwell's equations that describe electricity and magnetism. Magnetic Wiki field Book Mounir Gmati 40

Force on moving charges and current

Force on a charged particle

A charged particle moving in a B-field experiences a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz force, and is given by

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. Magnetic Wiki field Book Mounir Gmati 41

Magnetic force on charged particles and currents, shown in 3d. The electric current shown here is conventional, the real current would be a charge of –q flowing in exactly the opposite direction. Only one charge carrier is shown to prevent cluttering the diagram. r is the position of entry into the field B, r is 1 2 the exit. The vector l is the integral (sum) of all all infinitesimal vectors dr from r to r . 1 2

where F is the force, q is the electric charge of the particle, v is the instantaneous velocity of the particle, and B is the magnetic field (in teslas). The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field it will trace out a helical path in which the helix axis is parallel to the magnetic field and in which the speed of the particle will remain constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no work on an isolated charge. It can only do work indirectly, via the electric field generated by a changing magnetic field. It is often claimed that the magnetic force can do work to a non-elementary magnetic dipole, or to charged particles whose motion is constrained by other forces, but this is incorrect[26] because the work in those cases is performed by the electric forces of the charges deflected by the magnetic field.

Force on current-carrying wire The force on a current carrying wire is similar to that of a moving charge as expected since a charge carrying wire is a collection of moving charges. A current carrying wire feels a sideways force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the Laplace force. Consider a conductor of length l and area of cross section A and has charge q which is due to electric current i .If a conductor is placed in a magnetic field of induction B which makes an angle θ (theta) with the velocity of charges in the conductor which has i current Magnetic Wiki field Book Mounir Gmati 42

flowing in it. then force exerted due to small particle q is

then for n number of charges it has

then force exerted on the body is

but

that is

Direction of force

The direction of force on a charge or a current can be determined by a mnemonic known as the right-hand rule. See the figure on the left. Using the right hand and pointing the thumb in the direction of the moving positive charge or positive current and the fingers in the direction of the magnetic field the resulting force on the charge points outwards from the palm. The force on a negatively charged particle is in the opposite direction. The right-hand rule: Pointing the thumb of the right hand in the direction of the conventional current and the fingers in the direction If both the speed and the charge are reversed then the of the B-field the force on the current points out of the palm. The direction of the force remains the same. For that reason force is reversed for a negative charge. a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see Hall effect below.

An alternative mnemonic to the right hand rule is Fleming's left hand rule.

H and B inside and outside of magnetic materials The formulas derived for the magnetic field above are correct when dealing with the entire current. A magnetic material placed inside a magnetic field, though, generates its own bound current which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and the spin of the subatomic particles such as electrons that make up the material.) The H-field as defined above helps factor out this bound current; but in order to see how, it helps to introduce the concept of magnetization first. Magnetic Wiki field Book Mounir Gmati 43

Magnetization The magnetization field M represents how strongly a region of material is magnetized. For a uniform magnet, the magnetization is equal to its magnetic moment, m, divided by its volume. More generally, the magnetization of a region is defined as net magnetic dipole moment per unit volume of that region. Since the SI unit of magnetic moment is ampere-turn meter2, the SI unit of magnetization M is ampere-turn per meter which is 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 the region and is in the same direction as the local B-field it produces. Therefore, M field lines move from near the south pole of a magnet to near its north. Unlike B, magnetization only exists inside a magnetic material. Therefore, magnetization field lines begin and end near magnetic poles. The physically correct way to represent magnetization is to add all of the currents of the dipole moments that produce the magnetization. See Magnetic dipoles below and magnetic poles vs. atomic currents for more information. The resultant current is called bound current and is the source of the magnetic field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law: [27]

where the integral is a line integral over any closed loop and I is the 'bound current' enclosed by that closed loop. b It is also possible to model the magnetization in terms of magnetic charge in which magnetization begins at and ends at bound 'magnetic charges'. If a given region, therefore, has a net positive 'magnetic charge' then it will have more magnetic field lines entering it than leaving it. Mathematically this is equivalent to:

,

where the integral is a closed surface integral over the closed surface S and q is the 'magnetic charge' (in units of M magnetic flux) enclosed by S. (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because, like B inside a magnet, the magnetization field moves from south to north.

H-field and magnetic materials The H-field is defined as:

(definition of H in SI units)

With this definition, Ampere's law becomes:

where I represents the 'free current' enclosed by the loop so that the line integral of H does not depend at all on the f bound currents.[28] For the differential equivalent of this equation see Maxwell's equations. Ampere's law leads to the boundary condition

where K is the surface free current density.[29] f 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: Magnetic Wiki field Book Mounir Gmati 44

which does not depend on the free currents. The H-field, therefore, can be separated into two[30] independent parts:

where H is the applied magnetic field due only to the free currents and H is the demagnetizing field due only to the 0 d bound currents. The magnetic H-field, therefore, re-factors the bound current in terms of 'magnetic charges'. The H field lines loop only around 'free current' and, unlike the magnetic B field, begins and ends at near magnetic poles as well.

Magnetism Most materials respond to an applied B-field by producing their own magnetization M and therefore their own B-field. Typically, the response is very weak and exists only when the magnetic field is applied. The term 'magnetism' describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic phase of a material. Materials are divided into groups based upon their magnetic behavior: • Diamagnetic materials[31] produce a magnetization that opposes the magnetic field. • Paramagnetic materials[31] produce a magnetization in the same direction as the applied magnetic field. • Ferromagnetic materials and the closely related ferrimagnetic materials and antiferromagnetic materials[32] [33] can have a magnetization independent of an applied B-field with a complex relationship between the two fields. • Superconductors (and ferromagnetic superconductors)[34] [35] are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so 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 magnetic field such that:

where μ is a material dependent parameter called the permeability. In some cases the permeability may be a second rank tensor so that H may not point in the same direction as B. These relations between B and H are examples of constitutive equations. However, superconductors and ferromagnets have a more complex B to H relation, see magnetic hysteresis.

Energy stored in magnetic fields Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. For non-dispersive materials this same energy is released when the magnetic field is destroyed so that this energy can be modeled as being stored in the 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 μ . The above equation cannot be used for 0 nonlinear materials, though; a more general expression given below must be used. In general, the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δB is:

Once the relationship between H and B is known this equation is used to determine the work needed to reach a given magnetic state. For hysteretic materials such as ferromagnets and superconductors the work needed will also depend Magnetic Wiki field Book Mounir Gmati 45

on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above.

Electromagnetism: the relationship between magnetic and electric fields

Faraday's Law: Electric force due to a changing B-field A changing magnetic field, such as a magnet moving through a conducting coil, generates an electric field (and therefore tends to drive a current in the coil). This is known as Faraday's law and forms the basis of many electrical generators and electric motors. Mathematically, Faraday's law is:

where is the electromotive force (or EMF, the voltage generated around a closed loop) and Φ is the magnetic m flux—the product of the area times the magnetic field normal to that area. (This definition of magnetic flux is why B is 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 a coil produces a magnetic field that opposes the change in the magnetic field that induced it. This phenomenon is known as Lenz's Law. This integral formulation of Faraday's law can be converted[36] into a differential form, which applies under slightly different conditions. This form is covered as one of Maxwell's equations below.

Maxwell's correction to Ampère's Law: The magnetic field due to a changing electric field Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field. This fact is known as 'Maxwell's correction to Ampère's law'. Maxwell's correction to Ampère's Law bootstrap together with Faraday's law of induction to form electromagnetic waves, such as light. Thus, a changing electric 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 term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front. (The electric flux through an area is proportional to the area times the perpendicular part of the electric field.) This full Ampère law including the correction term is known as the Maxwell–Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used. The Maxwell 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. Magnetic Wiki field Book Mounir Gmati 46

Maxwell's equations Like all vector fields the B-field has two important mathematical properties that relates it to its sources. (For magnetic fields the sources are currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with the Lorentz force law form a complete description of classical electrodynamics including both electricity and magnetism. The first property is the divergence of a vector field A, ∇ · A which represents how A 'flows' outward from a given point. As discussed above, a B-field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of B is zero. (Such vector fields are called solenoidal vector fields.) This property is called Gauss's law for magnetism and is equivalent to the statement that there are no magnetic charges or magnetic monopoles. The electric field on the other hand begins and ends at electric charges so that 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 a given point. The result of the curl is called a 'circulation source'. The equations for the curl of B and of E are called the Ampère–Maxwell equation and Faraday's law respectively. They represent the differential forms of the integral equations given above. The complete set of Maxwell's equations then are:

Magnetic field, like all pseudovectors, changes sign when reflected in a mirror: When a loop of wire (black), carrying a current is reflected in a mirror (dotted line), the magnetic field it generates (blue) is not simply reflected in the mirror; rather, it is reflected and reversed.

where J = complete microscopic current density and ρ is the charge density. Technically, B is a pseudovector (also called an axial vector) due to being defined by a vector cross product. Because of the right-hand rule, a current-carrying loop viewed in a mirror results in a B vector that is both mirror-imaged and flipped in orientation, whereas an ordinary vector (e.g., velocity) is mirror-imaged only. (See diagram to right.) As discussed above, materials respond to an applied electric E field and an applied magnetic B field by producing their own internal 'bound' charge and current distributions that contribute to E and B but are difficult to calculate. To circumvent this problem the auxiliary H and D fields are defined so that Maxwell's equations can be re-factored in terms of the free current density J and free charge density ρ : f f Magnetic Wiki field Book Mounir Gmati 47

These equations are not any more general than the original equations (if the 'bound' charges and currents in the material are known'). They also need to be supplemented by the relationship between B and H as well as that between E and D. On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.

Electric and magnetic fields: different aspects of the same phenomenon According to the special theory of relativity, the partition of the electromagnetic force into separate electric and magnetic components is not fundamental, but varies with the observational frame of reference: An electric force perceived by one observer may be perceived by another (in a different frame of reference) as a magnetic force, or a mixture of electric and magnetic forces. Formally, special relativity combines the electric and magnetic fields into a rank-2 tensor, called the electromagnetic tensor. Changing reference frames mixes these components. This is analogous to the way that special relativity mixes space and time into spacetime, and mass, momentum and energy into four-momentum.

Magnetic vector potential In advanced topics such as quantum mechanics and relativity it is often easier to work with a potential formulation of electrodynamics 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[37] just as φ is interpreted 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 special relativity with little effort.[38] In relativity A together with φ forms the four-potential analogous to the four-momentum which combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler; further it can be easily modified to work with quantum mechanics. Magnetic Wiki field Book Mounir Gmati 48

Quantum electrodynamics In modern physics, the electromagnetic field is understood to be not a classical field, but rather a quantum field; it is represented 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 Quantum electrodynamics (QED),[39] which is incorporated into a more complete theory known as the "Standard Model of particle physics". In QED, the magnitude of the electromagnetic interactions between charged particles (and their antiparticles) is computed using perturbation theory; these rather complex formulas have a remarkable pictorial representation as Feynman diagrams in which virtual photons are exchanged. Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10−12 (and limited by experimental errors); for details see precision tests of QED. This makes QED one of the most accurate physical theories constructed thus far. All equations in this article are in the classical approximation, which is less accurate than the quantum description mentioned 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 The Earth's magnetic field is thought to be produced by convection currents in the outer liquid of Earth's core. The Dynamo theory proposes that these movements produce electric currents which, in turn, produce the magnetic field.[40] The presence of this field causes a compass, placed anywhere within it, to rotate so that the "north pole" of the magnet in the compass points roughly north, toward Earth's north magnetic pole. This is the traditional definition of the "north pole" of a magnet, although other equivalent definitions are also possible. One confusion that arises from this definition is that, if Earth itself is considered as a magnet, the south pole of that magnet would be the one nearer the north magnetic pole, and vice-versa[41] (opposite poles attract, so the north pole of the compass magnet is attracted to the south pole of Earth's interior magnet).

The north magnetic pole is so-named not because of the polarity of the field there but because of its geographical location. The north and south poles of a permanent magnet are so-called because they are "north-seeking" and "south-seeking", respectively.[42] The figure to the right is a sketch of Earth's magnetic field represented by field lines. For most locations, the magnetic field has a significant up/down component in addition to the North/South component. (There is also A sketch of Earth's magnetic field representing the source of the an East/West component; Earth's magnetic poles do field as 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 that not coincide exactly with Earth's geological pole.) The magnet is deep in Earth's interior below Earth's North Magnetic Pole. magnetic field can be visualised as a 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 Magnetic Wiki field Book Mounir Gmati 49

occurred 780,000 years ago.

Rotating magnetic fields The rotating magnetic field is a key principle in the operation of alternating-current motors. A permanent magnet in such a field rotates so as to maintain its alignment with the external field. This effect was conceptualized by Nikola Tesla, 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 AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents. This inequality would cause serious problems in standardization of the conductor size and so, in order to overcome it, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's electrical power supply systems. Because magnets degrade with time, synchronous motors use DC voltage fed rotor windings which allows the excitation 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 eddy currents 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 independently researched the concept. In 1888, Tesla gained U.S. Patent 381968 [43] for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.

Hall effect The charge carriers of a current carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field. The resultant voltage in that direction is proportional to the applied magnetic field. This is known as 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 the dominant charge carriers in materials such as (negative electrons or positive holes).

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

where is the magnetic flux in the circuit, is the magnetomotive force applied to

the circuit, and is the reluctance of the circuit. Here the reluctance is a quantity similar in nature to resistance for the flux. Using this analogy it is straight-forward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of circuit theory. Magnetic Wiki field Book Mounir Gmati 50

Magnetic field shape descriptions

• 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 solar plasma causes the meridional magnetic field to stretch into an azimuthal magnetic field, a process called the omega-effect. The reverse process is called the alpha-effect.[44] • A dipole magnetic field is one seen around a bar magnet or around a charged elementary particle with nonzero spin. • A quadrupole magnetic field is one seen, for example, between the poles of four bar magnets. The field strength grows linearly with the radial distance from its longitudinal axis. • A solenoidal magnetic field is similar to a dipole magnetic field, Schematic quadrupole magnet ("four-pole") except that a solid bar magnet is replaced by a hollow magnetic field. There are four steel pole tips, two opposing magnetic north poles and two opposing electromagnetic coil magnet. magnetic south poles. • A toroidal magnetic field occurs in a doughnut-shaped coil, the electric 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).[45] • A helical magnetic field is corkscrew-shaped, and sometimes seen in space plasmas such as the Orion Molecular Cloud.[46]

Magnetic dipoles

The magnetic field of a magnetic dipole is depicted on the right. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current I and an area a. Such a current loop has a magnetic moment of:

where the direction of m is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area a has been reduced to zero and its current I increased to infinity such that the product m = Ia is finite. In this model it is easy to see the connection between angular momentum and magnetic moment which Magnetic field lines around a ”magnetostatic dipole” pointing to the right. is the basis of the Einstein-de Haas effect "rotation by magnetization" and its inverse, the Barnett effect or "magnetization by rotation".[47] Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example.

It is sometimes useful to model the magnetic dipole similar to the electric dipole with two equal but opposite magnetic 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 between electricity and magnetism. Further, as discussed above it fails to explain the inherent connection between angular momentum and magnetism. Magnetic Wiki field Book Mounir Gmati 51

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] By the definition of magnetization, in this model, and in analogy to the physics of springs, the work done per unit volume, in stretching and twisting the bonds between magnetic charge to increment the magnetization by μ δM is W = H · μ δM. In this model, B = μ (H + M ) is an 0 0 0 effective magnetization which includes the H-field term to account for the energy of setting up the magnetic field in a vacuum. Therefore the total energy density increment needed to increment the magnetic field is W = H · δB. [4] It is a remarkable fact that from the 'outside' the field of a dipole of magnetic charge has the exact same form as that of an elementary current loop called a magnetic dipole. It is therefore only for the physics of magnetism 'inside' of magnetic material that the two models differ.

[5] Electromagnetics, by Rothwell and Cloud, p23 (http:/ / books. google. com/ books?id=jCqv1UygjA4C& pg=PA23) [6] R.P. Feynman, R.B. Leighton, M. Sands (1963). The Feynman Lectures on Physics, volume 2. [7] Edward Purcell, in Electricity and Magnetism, McGraw-Hill, 1963, writes, Even some modern writers who treat B as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by H. This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling B the magnetic field. As for H, although other names have been invented for it, we shall call it "the field H" or even "the magnetic field H." In a similar vein, M Gerloch (1983). Magnetism and

Ligand-field Analysis (http:/ / books. google. com/ ?id=Ovo8AAAAIAAJ& pg=PA110). Cambridge University Press. p. 110. ISBN 0521249392. . says: "So we may think of both B and H as magnetic fields, but drop the word 'magnetic' from H so as to maintain the distinction ... As Purcell points out, 'it is only the names that give trouble, not the symbols'." [8] This can be seen from the magnetic part of the Lorentz force law F = (qvB). mag

[9] Magnetic Field Strength Converter (http:/ / www. unitconversion. org/ unit_converter/ magnetic-field-strength. html), UnitConversion.org.

[10] "Gravity Probe B Executive Summary" (http:/ / www. nasa. gov/ pdf/ 168808main_gp-b_pfar_cvr-pref-execsum. pdf). pp. 10,21. .

[11] "With record magnetic fields to the 21st Century" (http:/ / ieeexplore. ieee. org/ xpl/ freeabs_all. jsp?arnumber=823621). IEEE Xplore. .

[12] Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). " Magnetars (http:/ / solomon. as. utexas. edu/ ~duncan/ sciam. pdf)". Scientific American; Page 36. [13] 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. [14] 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 magnetic fields of 'elementary parts' having magnetic field lines that do form closed curves or extend to infinity. [15] To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north pole of the magnet since magnets stacked on each other point in the same direction. [16] As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total 'number' of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the main text are used instead. [17] Two experiments produced candidate events that were initially interpreted as monopoles, but these are now regarded to be inconclusive. For details and references, see magnetic monopole. [18] 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. [19] Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 255–8. ISBN 0-13-805326-X. OCLC 40251748. [20] Either B or H may be used for the magnetic field outside of the magnet.

[21] 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 0387989730. . [22] Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. p. 438. ISBN 0-13-805326-X. OCLC 40251748. [23] 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. [24] Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 222–225. ISBN 0-13-805326-X. OCLC 40251748. [25] 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.)

[26] 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. . [27] Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 266–8. ISBN 0-13-805326-X. OCLC 40251748.

[28] 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 0486622630. . Magnetic Wiki field Book Mounir Gmati 52

[29] David Griffiths. Introduction to Electrodynamics (3rd 1999 ed.). p. 332. [30] A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below.

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

[32] 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 0198517769. .

[33] 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 0198508085. .

[34] 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 3540732527. .

[35] 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 1590338618. . [36] 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 a distance dℓ based upon the Lorentz force law. In the case where the bounding surface is stationary, the Kelvin–Stokes theorem can be used to show this equation is equivalent to the Maxwell–Faraday equation. [37] 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. [38] Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. p. 422. ISBN 0-13-805326-X. OCLC 40251748. [39] 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.

[40] 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): p.266. .

[41] College Physics, Volume 10, by Serway, Vuille, and Faughn, page 628 weblink (http:/ / books. google. com/ books?id=CX0u0mIOZ44C& pg=PT660). "the geographic North Pole of Earth corresponds to a magnetic south pole, and the geographic South Pole of Earth corresponds to a magnetic north pole".

[42] Kurtus, Ron (2004). "Magnets" (http:/ / www. school-for-champions. com/ science/ magnets. htm). School for champions: Physics topics. . Retrieved 17 July 2010.

[43] http:/ / www. google. com/ patents?vid=381968

[44] The Solar Dynamo (http:/ / www. cora. nwra. com/ ~werne/ eos/ text/ dynamo. html), retrieved Sep 15, 2007.

[45] 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 Oct 2008

[46] 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 Oct 2008 [47] (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 0471477419. .

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. • 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: Magnetic Wiki field Book Mounir Gmati 53

4111–4117. • 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. • 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.

External links

Information Rotating magnetic fields

• Crowell, B., " Electromagnetism (http:/ / www. lightandmatter. com/ • " Rotating magnetic fields (http:/ / www. tpub. com/ neets/

html_books/ 0sn/ ch11/ ch11. html)". book5/ 18a. htm)". Integrated Publishing.

• Nave, R., " Magnetic Field (http:/ / hyperphysics. phy-astr. gsu. edu/ hbase/ •"Introduction to Generators and Motors", rotating magnetic

magnetic/ magfie. html)". HyperPhysics. field (http:/ / www. tpub. com/ content/ neets/ 14177/ css/

•"Magnetism", The Magnetic Field (http:/ / theory. uwinnipeg. ca/ physics/ 14177_87. htm). Integrated Publishing.

mag/ node2. html#SECTION00110000000000000000). theory.uwinnipeg.ca. • " Induction Motor – Rotating Fields (http:/ / www. egr. msu.

• Hoadley, Rick, " What do magnetic fields look like (http:/ / my. execpc. com/ edu/ ~jurkovi4/ Experiment4. pdf)". (dead link)

~rhoadley/ magfield. htm)?" 17 July 2005. Diagrams Field density • McCulloch, Malcolm,"A2: Electrical Power and Machines",

• Oppelt, Arnulf (2006-11-02). "magnetic field strength" (http:/ / searchsmb. Rotating magnetic field (http:/ / www. eng. ox. ac. uk/

techtarget. com/ sDefinition/ 0,290660,sid44_gci763586,00. html). Retrieved ~epgmdm/ A2/ img89. htm). eng.ox.ac.uk. 2007-06-04. •"AC Motor Theory" Figure 2 Rotating Magnetic Field (http:/

• "magnetic field strength converter" (http:/ / www. unitconversion. org/ / www. tpub. com/ content/ doe/ h1011v4/ css/ h1011v4_23.

unit_converter/ magnetic-field-strength. html). Retrieved 2007-06-04. 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. Flux pinning Wiki Book Mounir Gmati 54 Flux pinning

Flux pinning is the phenomenon that magnetic flux lines do not move (become trapped, or "pinned") in spite of the Lorentz force acting on them inside a current-carrying Type II superconductor. The phenomenon cannot occur in Type I superconductors, since these cannot be penetrated by magnetic fields (Meissner–Ochsenfeld effect). Flux pinning is only possible when there are defects in the crystalline structure of the superconductor (usually resulting from grain boundaries or impurities).

Importance of flux pinning Flux pinning is desirable in high-temperature ceramic superconductors to prevent "flux creep", which can create a pseudo-resistance and depress both critical current density and critical field. Degradation of a high-temperature superconductor's properties due to flux creep is a limiting factor in the use of these superconductors. SQUID magnetometers suffer reduced precision in a certain range of applied field due to flux creep in the superconducting magnet used to bias the sample, and the maximum field strength of high-temperature superconducting magnets is drastically reduced by the depression in critical field.

References • Future Science [1] introduction to high-temperature superconductors. • American Magnetics [2] tutorial on magnetic field exclusion and flux pinning in superconductors. • Cern Lhc documentation [3] Stability of superconductors.

Other sources • Flux-Pinning of Bi Sr CaCu O High T Superconducting Tapes Utilizing (Sr,Ca) Cu O and 2 2 2 (8 + Delta) c 14 24 (41 + Delta) Sr CaAl O Defects (T. Haugan; et al. AFB OH Propulsion Directorate. Air Force Research Lab 2 2 6 Wright-Patterson. Oct 2003)

References

[1] http:/ / www. futurescience. com/ scintro. html

[2] http:/ / www. americanmagnetics. com/ supercon. php

[3] http:/ / quench-analysis. web. cern. ch/ quench-analysis/ phd-fs-html/ node41. html Magnetic Wiki levitation Book Mounir Gmati 55 Magnetic levitation

Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields. Magnetic pressure is used to counteract the effects of the gravitational and any other accelerations. Earnshaw's theorem proves that using only static ferromagnetism it is impossible to stably levitate against gravity, but servomechanisms, the use of diamagnetic materials, superconduction, or systems involving eddy currents permit this to occur. In some cases the lifting force is provided by magnetic levitation, but there is a mechanical support bearing little load that provides stability. This is termed pseudo-levitation. Magnetic levitation is used for maglev trains, magnetic bearings and for product display purposes.

Lift Levitating pyrolytic carbon

Magnetic materials and systems are able to attract or press each other apart or together with a force dependent on the magnetic field and the area of the magnets, and a magnetic pressure can then be defined. The magnetic pressure of a magnetic field on a superconductor can be calculated by:

where is the force per unit area in pascals, is the magnetic field just above the superconductor in teslas, and = 4π×10−7 N·A−2 is the permeability of the vacuum.[1]

Stability Static stability means that any small displacement away from a stable equilibrium causes a net force to push it back to the equilibrium point. Earnshaw's theorem proved conclusively that it is not possible to levitate stably using only static, macroscopic, paramagnetic fields. The forces acting on any paramagnetic object in any combinations of gravitational, electrostatic, and magnetostatic fields will make the object's position, at best, unstable along at least one axis, and it can be unstable equilibrium along all axes. However, several possibilities exist to make levitation viable, for example, the use of electronic stabilization or diamagnetic materials (since relative magnetic permeability is less than one[2] ); it can be shown that diamagnetic materials are stable along at least one axis, and can be stable along all axes. Conductors can have a relative permeability to alternating magnetic fields of below one, so some configurations using simple AC driven electromagnets are self stable. Dynamic stability occurs when the levitation system is able to damp out any vibration-like motion that may occur. Magnetic Wiki levitation Book Mounir Gmati 56

Methods For successful levitation and control of all 6 axes (3 spatial and 3 rotational) a combination of permanent magnets and electromagnets or diamagnets or superconductors as well as attractive and repulsive fields can be used. From Earnshaw's theorem at least one stable axis must be present for the system to levitate successfully, but the other axes can be stabilised using ferromagnetism. The primary ones used in maglev trains are servo-stabilized electromagnetic suspension (EMS), electrodynamic suspension (EDS), and experimentally, Inductrack.

Mechanical constraint (pseudo-levitation)

With a small amount of mechanical constraint for stability, pseudo-levitation is relatively straightforwardly achieved. If two magnets are mechanically constrained along a single vertical axis, for example, and arranged to repel each other strongly, this will act to levitate one of the magnets above the other. Another geometry is where the magnets are attracted, but constrained from touching by a tensile member, such as a string or cable. Another example is the Zippe-type centrifuge where a cylinder is suspended under an attractive magnet, and stabilized by a needle bearing from below.

Mechanical constraint (in this case the lateral restrictions created by a box) can permit pseudo-levitation of permanent magnets

Diamagnetism Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the magnetic dipole moment. According to Lenz's law, this opposes the external field. Diamagnets are materials with a magnetic permeability less than μ0 (a relative permeability less than 1). Consequently, diamagnetism is a form of magnetism that is only exhibited by a substance in the presence of an externally applied magnetic field. It is generally quite a weak effect in most materials, although superconductors exhibit a strong effect. Diamagnetic materials cause lines of magnetic flux to curve away from the material, and superconductors can exclude them completely (except for a very thin layer at the surface). Magnetic Wiki levitation Book Mounir Gmati 57

Direct diamagnetic levitation

A substance that is diamagnetic repels a magnetic field. All materials have diamagnetic properties, but the effect is very weak, and is usually overcome by the object's paramagnetic or ferromagnetic properties, which act in the opposite manner. Any material in which the diamagnetic component is strongest will be repelled by a magnet.

Earnshaw's theorem does not apply to diamagnets. These behave in the opposite manner to normal magnets owing to their relative permeability of μ < 1 (i.e. negative magnetic susceptibility). r Diamagnetic levitation can be used to levitate very light pieces of pyrolytic graphite or bismuth above a moderately strong

permanent magnet. As water is predominantly diamagnetic, this A live frog levitates inside a 32 mm diameter vertical technique has been used to levitate water droplets and even live bore of a Bitter solenoid in a magnetic field of about 16 [3] animals, such as a grasshopper, frog and a mouse. However, the teslas at the High Field Magnet Laboratory of the Radboud University in Nijmegen the Netherlands. magnetic fields required for this are very high, typically in the [4] Direct link to video range of 16 teslas, and therefore create significant problems if ferromagnetic materials are nearby.

The minimum criterion for diamagnetic levitation is , where:

• is the magnetic susceptibility • is the density of the material • is the local gravitational acceleration (−9.8 m/s2 on Earth) • is the permeability of free space • is the magnetic field

• is the rate of change of the magnetic field along the vertical axis.

Assuming ideal conditions along the z-direction of solenoid magnet:

• Water levitates at

• Graphite levitates at Magnetic Wiki levitation Book Mounir Gmati 58

Diamagnetically-stabilized levitation A permanent magnet can be stably suspended by various configurations of strong permanent magnets and strong diamagnets. When using superconducting magnets, the levitation of a permanent magnet can even be stabilized by the small diamagnetism of water in human fingers.[5]

Superconductors

Superconductors may be considered perfect diamagnets (μ = 0), as r well as the property they have of completely expelling magnetic fields due to the Meissner effect when the superconductivity initially forms. The levitation of the magnet is further stabilized due to flux pinning within the superconductor; this tends to stop the superconductor leaving the magnetic field, even if the levitated system is inverted.

These principles are exploited by EDS (Electrodynamic Suspension), superconducting bearings, flywheels, etc. In trains, a very strong magnetic field is required to levitate a massive train, the JR–Maglev have superconducting magnetic coils. JR–Maglev levitation is not by Meissner effect.

Rotational stabilization

A magnet can be levitated against gravity when gyroscopically stabilized by spinning it in a toroidal field created by a base ring of A superconductor levitating a permanent magnet magnet(s). However, it will only remain stable until the rate of precession slows below a critical threshold—the region of stability is quite narrow both spatially and in the required rate of precession. The first discovery of this phenomenon was by Roy Harrigan, a Vermont inventor who patented a levitation device in 1983 based upon it.[6] Several devices using rotational stabilization (such as the popular Levitron toy) have been developed citing this patent. Non-commercial devices have been created for university research laboratories, generally using magnets too powerful for safe public interaction.

Servomechanisms

The attraction from a fixed strength magnet decreases with increased distance, and increases at closer distances. This is unstable. For a stable system, the opposite is needed, variations from a stable position should push it back to the target position. Stable magnetic levitation can be achieved by measuring the position and speed of the object being levitated, and using a feedback loop which continuously adjusts one or more electromagnets to correct the object's motion, thus forming a servomechanism.

Many systems use magnetic attraction pulling upwards against gravity The Transrapid system uses servomechanisms to for these kinds of systems as this gives some inherent lateral stability, pull the train up from underneath the track and but some use a combination of magnetic attraction and magnetic maintains a constant gap while travelling at high speed repulsion to push upwards.

Either system represents examples of ElectroMagnetic Suspension (EMS). For a very simple example, some tabletop levitation demonstrations use this principle, and the object cuts a beam of light to measure the position of the object. Magnetic Wiki levitation Book Mounir Gmati 59

The electromagnet is above the object being levitated; the electromagnet is turned off whenever the object gets too close, and turned back on when it falls further away. Such a simple system is not very robust; far more effective control systems exist, but this illustrates the basic idea. EMS magnetic levitation trains are based on this kind of levitation: The train wraps around the track, and is pulled upwards from below. The servo controls keep it safely at a constant distance from the track.

Induced currents These schemes work due to repulsion due to Lenz's law. When a conductor is presented with a time-varying magnetic field electrical currents in the conductor are set up which create a magnetic field that causes a repulsive effect.

Relative motion between conductors and magnets If one moves a base made of a very good electrical conductor such as copper, aluminium or silver close to a magnet, an (eddy) current will be induced in the conductor that will oppose the changes in the field and create an opposite field that will repel the magnet (Lenz's law). At a sufficiently high rate of movement, a suspended magnet will levitate on the metal, or vice versa with suspended metal. Litz wire made of wire thinner than the skin depth for the frequencies seen by the metal works much more efficiently than solid conductors. An especially technologically-interesting case of this comes when one uses a Halbach array instead of a single pole permanent magnet, as this almost doubles the field strength, which in turn almost doubles the strength of the eddy currents. The net effect is to more than triple the lift force. Using two opposed Halbach arrays increases the field even further.[7] Halbach arrays are also well-suited to magnetic levitation and stabilisation of gyroscopes and electric motor and generator spindles.

Oscillating electromagnetic fields A conductor can be levitated above an electromagnet (or vice versa) with an alternating current flowing through it. This causes any regular conductor to behave like a diamagnet, due to the eddy currents generated in the conductor.[8] [9] Since the eddy currents create their own fields which oppose the magnetic field, the conductive object is repelled from the electromagnet, and most of the field lines of the magnetic field will no longer penetrate the conductive object. This effect requires non-ferromagnetic but highly conductive materials like aluminium or copper, as the ferromagnetic ones are also strongly attracted to the electromagnet (although at high frequencies the field can still be expelled) and tend to have a higher resistivity giving lower eddy currents. Again, litz wire gives the best results. The effect can be used for stunts such as levitating a telephone book by concealing an aluminium plate within it. At high frequencies (a few tens of kilohertz or so) and kilowatt powers small quantities of metals can be levitated and melted using levitation melting without the risk of the metal being contaminated by the crucible.[10] One source of oscillating magnetic field that is used is the linear induction motor. This can be used to levitate as well as provide propulsion. Magnetic Wiki levitation Book Mounir Gmati 60

Strong focusing Earnshaw's theory strictly only applies to static fields. Alternating magnetic fields, even purely alternating attractive fields,[11] can induce stability and confine a trajectory through a magnetic field to give a levitation effect. This is used in particle accelerators to confine and lift charged particles, and has been proposed for maglev trains also.[11]

Dynamic stability Magnetic fields are conservative forces and therefore in principle have no built-in damping, and in practice many of the levitation schemes are under-damped and in some cases negatively damped.[12] This can permit vibration modes to exist that can cause the item to leave the stable region. Damping of motion is done in a number of ways: • external mechanical damping (in the support), such as dashpots, air drag etc. • eddy current damping (conductive metal influenced by field) • tuned mass dampers in the levitated object • electromagnets controlled by electronics

Difficulties Most of the levitation techniques have various complexities. • The power requirements of electromagnets increase rapidly with load-bearing capacity, which also necessitates relative increases in conductor and cooling equipment mass and volume. • Superconductors require very low temperatures to operate, often helium cooling is employed.

Uses

Maglev transportation Maglev, or magnetic levitation, is a system of transportation that suspends, guides and propels vehicles, predominantly trains, using magnetic levitation from a very large number of magnets for lift and propulsion. This method has the potential to be faster, quieter and smoother than wheeled mass transit systems. The technology has the potential to exceed 6,400 km/h (4,000 mi/h) if deployed in an evacuated tunnel.[13] If not deployed in an evacuated tube the power needed for levitation is usually not a particularly large percentage and most of the power needed is used to overcome air drag, as with any other high speed train. The highest recorded speed of a maglev train is 581 kilometers per hour (361 mph), achieved in Japan in 2003,[14] 6 km/h faster than the conventional TGV speed record. This is slower than many aircraft, since aircraft can fly at far higher altitudes where air drag is lower, thus high speeds are more readily attained. Magnetic Wiki levitation Book Mounir Gmati 61

Magnetic bearings • Magnetic bearings • Flywheels • Centrifuges • Magnetic ring spinning

Levitation melting Electromagnetic levitation (EML), patented by Muck in 1923 [15] , is one of the oldest levitation techniques used for containerless experiments.[16] The technique enables the levitation of an object using electromagnets. A typical EML coil has reversed winding of upper and lower sections energized by a radio frequency power supply.

History This list is incomplete. • 1839 Earnshaw's theorem showed electrostatic levitation was impossible, later theorem was extended to magnetostatic levitation by others • 1912 Emile Bachelet awarded a patent in March 1912 for his “levitating transmitting apparatus” (patent no. 1,020,942) for electromagnetic suspension system • 1933 Walter Meissner and Robert Ochsenfeld (the Meissner effect) • 1934 Hermann Kemper “monorail vehicle with no wheels attached.” Reich Patent number 643316 • 1939 Braunbeck’s extension showed that magnetic levitation is possible with diamagnetic materials • 1939 Bedford, Peer, and Tonks aluminum plate placed on two concentric cylindrical coils shows 6-axis stable levitation.[17] • 1961 James R. Powell and BNL colleague Gordon Danby electrodynamic levitation using superconducting magnets • 1970s Spin stabilized magnetic levitation Roy M. Harrigan • 1974 Magnetic river Eric Laithwaite and others • 1979 transrapid train carried passengers • 1984 Low speed maglev shuttle in Birmingham Eric Laithwaite and others • 1999 Inductrack permanent magnet electrodynamic levitation (General Atomics) • 2000 Diamagnetically levitated live frog Andre Geim

References

[1] Lecture 19 MIT 8.02 Electricity and Magnetism, Spring 2002 [2] Braunbeck, W. Free suspension of bodies in electric and magnetic fields, Zeitschrift für Physik, 112, 11, pp753-763 (1939)

[3] http:/ / RU. nl/ HFML

[4] http:/ / web. Archive. org/ web/ 20070211113825/ http:/ / www. HFML. RU. nl/ pics/ Movies/ frog. mpg

[5] Diamagnetically stabilized magnet levitation (http:/ / netti. nic. fi/ ~054028/ images/ LeviTheory. pdf)

[6] US patent 4382245 (http:/ / worldwide. espacenet. com/ textdoc?DB=EPODOC& IDX=US4382245), Roy M. Harrigan, "Levitation device", issued 1983-05-03

[7] (https:/ / www. llnl. gov/ str/ November03/ Post. html)

[8] Eddy current magnetic levitation, models and experiments by Marc T. Thompson (http:/ / www. classictesla. com/ download/

ieee_potentials_2000. pdf)

[9] Levitated Ball-Levitating a 1 cm aluminum sphere (http:/ / sprott. physics. wisc. edu/ demobook/ chapter5. htm) [10] "Magnetic levitation of liquid metals", Journal of Fluid Mechanics 117, pages 27-43, by A. J. Mestel [11] Attractive levitation for high-speed ground transport with largeguideway clearance and alternating-gradient stabilization Hull, J.R.

Magnetics, IEEE Transactions on Volume 25, Issue 5, Sep 1989 Page(s):3272 - 3274 Digital Object Identifier 10.1109/20.42275 (http:/ /

ieeexplore. ieee. org/ Xplore/ login. jsp?url=/ iel1/ 20/ 1621/ 00042275. pdf?tp=& isnumber=1621& arnumber=42275& type=ref) [12] A Review of Dynamic Stability of Repulsive-Force Maglev Suspension Systems- Y. Cai and D.M.Rote

[13] http:/ / www. popsci. com/ scitech/ article/ 2004-04/ trans-atlantic-maglev Magnetic Wiki levitation Book Mounir Gmati 62

[14] "Japan's maglev train sets speed record" (http:/ / www. theglobeandmail. com/ servlet/ story/ RTGAM. 20031202. gtmaglevdec2/ BNStory/

International/ ). CTVglobemedia Publishing Inc.. 2003-12-02. . Retrieved 2009-02-16. [15] O. Muck. German patent no. 42204 (Oct. 30, 1923) [16] Paul C. Nordine, J. K. Richard Weber, and Johan G. Abadie (2000), "Properties of high-temperature melts using levitation", Pure and Applied Chemistry 72: 2127–2136, doi:10.1351/pac200072112127 [17] linear Electric Machines- A Personal View ERIC R. LAITHWAITE, FELLOW, IEEE, PROCEEDINGS OF THE IEEE, VOL. 63, NO. 2, FEBRUARY 1975

External links

• Maglev Trains (http:/ / www. magnet. fsu. edu/ education/ community/ slideshows/ maglev/ index. html) Audio slideshow from the National High Magnetic Field Laboratory discusses magnetic levitation, the Meissner Effect, magnetic flux trapping and superconductivity

• Magnetic Levitation - Science is Fun (http:/ / www. levitationfun. com/ index. html)

• Magnetic (superconducting) levitation experiment (YouTube) (http:/ / www. youtube. com/

watch?v=nWTSzBWEsms& feature=related)

• Maglev video gallery (http:/ / users. bigpond. net. au/ com/ maglevvideogallery/ )

• How can you magnetically levitate objects? (http:/ / my. execpc. com/ ~rhoadley/ maglev. htm)

• Levitated aluminum ball (oscillating field) (http:/ / sprott. physics. wisc. edu/ demobook/ chapter5. htm)

• Instructions to build an optically triggered feedback maglev demonstration (http:/ / www. coilgun. info/ levitation/

home. htm)

• Videos of diamagnetically levitated objects, including frogs and grasshoppers (http:/ / www. hfml. sci. kun. nl/

levitation-movies. html)

• Larry Spring's Mendocino Brushless Magnetic Levitation Solar Motor (http:/ / www. larryspring. com/

class_motors. html)

• A Classroom Demonstration of Levitation... (http:/ / . org/ abs/ 0803. 3090)

• 25kg MAGLEV suspension setup (http:/ / www. youtube. com/ watch?v=Y_WG4YStMxs& feature=related)

• 25kg MAGLEV suspension control via Classical control strategy (http:/ / www. youtube. com/ watch?v=kXodf7WKiFs)

• 25kg MAGLEV suspension via State feedback control strategy (http:/ / www. youtube. com/

watch?v=TsgoF13KvYk& feature=related)

• Frogs levitate in a strong enough magnetic field (http:/ / www. physics. org/ facts/ frog-really. asp) Article SourcesWiki Book and Contributors Mounir Gmati 63 Article Sources and Contributors

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