Magnetic field

In Physical, the magnetic field is a size characterized by the data of an intensity and a direction, defined in any point of space, and determined by the position and the orientation of Aimant S, electromagnet S and the displacement of electric charges. The presence of this field results in the existence of a force acting on the electric charges moving (known as Force of Lorentz), and various effects affecting certain materials (Paramagnétisme, Diamagnétisme or Ferromagnétisme according to the cases). The size which determines the interaction between a Matériau and a magnetic field is the magnetic Susceptibilité.

The magnetic field forms, with the Electric field the two components of the electromagnetic Champ describes by the electromagnetism. Waves of fields electric and magnetic frays can be propagated freely in space, and the majority of materials. These waves are called electromagnetic waves, and correspond to all the manifestations of the Lumière, in all the fields of Wavelength (Ondes radio, field Micro-onde, Infrarouge, visible Domaine, Ultraviolet, X-rays and Gamma rays). The discipline which studies the static magnetic fields (not depending on the Temps) is the Magnétostatique.

The applications of the control of this field are numerous, even in the everyday life: in addition to the fact that this one is a component of the Lumière, he explains the attraction of the Aimant S, the orientation of the Boussole S and allows inter alia the construction of Alternateur S and electrical motors. The storage of information on magnetic bands or hard drives is done using magnetic fields. Magnetic fields of very strong intensity are used in the particle accelerator or the Tokamak S to focus a beam of very energy particles with an aim of making them enter in collision. The magnetic fields are also omnipresent in Astronomie, where they are in the beginning many phenomena like the Synchrotron radiation and the Rayonnement of curve, as well as the formation of jet S in the areas where one observes a Disque of accretion. The synchrotron radiation is also abundantly used in many industrial applications.

Mathematically, the magnetic field describes by a field pseudo vectorial, which approaches a vector Field by several aspects, but has some subtleties to the level of the symmetry S. the equations describing the evolution of the magnetic field are called Maxwell's equations, in the honor of James Clerk Maxwell which finalized them in 1873. It is however Albert Einstein which in 1905 proposed of it the first the most coherent vision, within the framework of the restricted Relativité which it had just discovered and which in is indissociable.

History

Manifestations of the magnetic field

See also: Magnetism, Electromagnetism

In traditional physics, the magnetic fields result from electric currents. At the microscopic level, a electron of “orbits” around an atomic nucleus can be seen as a tiny loop of current, generating a weak magnetic field and behaving like a magnetic Dipôle. According to the properties of materials, these microscopic magnetic structures will give place to primarily three types of phenomena:

In certain cases, the field generated by electrons of close atoms present a certain tendency to align the ones compared to the others, a macroscopic magnetic field, i.e. a spontaneous Aimantation, is likely to appear. It is the phenomenon of Ferromagnétisme, explaining the existence of permanent magnets. It is possible to destroy the magnetic field of a magnet by heating it beyond a certain temperature. The thermal Agitation generated by the heating breaks the interactions between close atoms which were responsible for the alignment of the atomic magnetic fields. In practice, the phenomenon of ferromagnetism disappears beyond from a certain temperature called Température of Curie. It is of 770 degrees Celsius for the Fer.

In the absence of ferromagnetism, or at a too high temperature so that this one appears, the presence of an external magnetic field can lead the microscopic fields to be aligned in the direction of the field. This phenomenon is called Paramagnétisme. The transition between the ferromagnetic state and the paramagnetic state is made via a Transition from phase known as of second order (i.e. magnetization tends continuously towards 0 as the temperature approaches the temperature of Curie, but that its derivative compared to the temperature diverges with the transition). The first mathematical model making it possible to reproduce such a behavior is called the Modèle of Ising, of which the resolution, considered as a mathematical feat of ingenuity, was carried out by the Nobel Prize of chemistry Lars Onsager in 1944.

Contrary, certain materials tend to react by aligning their microscopic magnetic fields in an antiparallel way with the field, i.e. endeavouring to decrease the imposed magnetic field of outside. Such a phenomenon is called Diamagnétisme.

Electric currents

All Electric current generates a magnetic field, which showed the historical experiment of Ørsted.
The presence of a current thus makes it possible to locally influence the magnetic field, it is the principle of the electromagnet S. This magnetic field is all the more intense as the current is. Reciprocally, a variable magnetic field is likely to generate an electric current. It is the principle of the magnetic Induction which all use the electric machines.

Magnetic fields of planets

See also: Magnetosphere, Terrestrial magnetic field, Polar lights

The Ground, like the majority of the Planet S of the Solar system, has a magnetic field. This Terrestrial magnetic field - which protects the Earth by deviating the particles charged resulting from the Sun in an area called Magnétosphère - is mainly of internal origin. It is supposed that it is resulting from effects of Convection of the matter located in the external core of the Earth, mainly made up of Fer and of Nickel liquid. In particular, of the current (although very weak), traversing the core would induce this magnetic field, by a process called $dynamo effect.
The median value of the terrestrial magnetic field is approximately 0,5 gauss (either 5.10 -5 T). The Terrestrial magnetic field fluctuates during time: its direction and its intensity are not constant. Moreover, it is not homogeneous in any point of the sphere.

In particular, the magnetic fields of the planets Jupiter and Saturn, most intense after that of the Sun are studied currently much so in particular including the shift between the orientation of the magnetic field and the axis of rotation of planet, like its variations. The measurement of the magnetic field of Saturn is one of the objectives of the Mission Cassini-Huygens, while that of Jupiter will be studied by the probe JUNO. The origin of these fields is supposed to be related to the movements of the core of metal Hydrogène that they shelter.
On the level of the magnetic poles of these planets, the field tends to guide the charged particles, resulting for example from the solar wind. Those, very energy, interact sometimes with the atmosphere of planet: it is what one can observe in the form of the polar lights.

Magnetic monopolies

See also: magnetic Monopoly

One of the basic differences between the Electric field and the magnetic field is that one observes in the nature of the particles having a electric Charge, whereas one observes neither particle nor object having a magnetic Charge. In practice that results in the absence of configurations having a purely radial magnetic field, which mathematically corresponds to the fact that the magnetic field is of null divergence.
In particular, any magnet has a north pole and a magnetic south pole. If this magnet into two is broken, one finds oneself with two magnets having each one a north pole and a magnetic south pole. Mathematically, this property results in the fact that the divergence of the magnetic field is null, property formalized by one of the Maxwell's equations. Hypothetical objects not having that only one magnetic pole are called magnetic monopolies.
On the other hand, within the framework of the quantum electrodynamic , such objects appear in the resolution of the equation of Dirac: they are the monopolies of Dirac. In the theory of Yang-Millets, one utilizes a Monopôle of 'T Hooft-Polyakov.

Relativistic origin

See also: Transformations of Lorentz of the electromagnetic field

In 1905, Albert Einstein showed how the magnetic field appears, like one of the relativistic aspects of the Electric field, more precisely within the framework of the restricted Relativité.
It is presented in the form of a result of the Lorentzian transformation of an electric field of a first reference frame to a second moving relative.
When an electric charge moves, the electric field generated by this load is not perceived any more by an observer at rest as with spherical symmetry, because of the dilation of time predicted by relativity. One must then employ the transformations of Lorentz to calculate the effect of this load on the observer, which gives a component of the field which acts only on the loads moving: what one calls “magnetic field”.
One can thus describe the magnetic fields and electric like two aspects of the same physical object, represented in theory of relativity restricted by a Tenseur of row 2.

Units and orders of magnitude

See also: Intensity of magnetic field

The modern unit used to quantify the intensity of the magnetic field is the Tesla, defined in 1960. It is a derived unit system IF. One defines a Tesla by a magnetic flow of Induction of a weber per square meter:

1 T = 1 Wb · m -2 = 1 kg · S -2 · has -1 = 1 NR ·With -1 ·m -1 = 1 kg·S -1 · C -1 .

For various historical reasons going back to work of Charles of Coulomb, certain authors prefer to use units out of the system IF, like the gauss or the gamma. One a:

1 Tesla = 10.000 gauss;

1 Tesla = 1.000.000 gamma.

Lastly, one uses also sometimes the oersted, in particular to quantify the “force” of the natural magnets, of which the equivalent IF is the amp per A.m^-1 meter by the relation:

$1 \, \ mathrm {Oe} = \ frac {10^3} {4 \ pi} \ mathrm {A.m^ {- 1}}$ .

In interplanetary space, the magnetic field is included/understood between 10 -10 and 10 -8 T. magnetic fields with more large scales, for example within the Milky Way are also measured, via the phenomenon of rotation of Faraday, in particular thanks to the observation of the Pulsar S. the origin and the galactic evolution of the magnetic fields on the scales S and beyond is at present (2007) an open problem in Astrophysique. The star S, following the example Planet S, have also a magnetic field, which can be highlighted by Spectroscopie (Effet Zeeman). A star at the end of the lifetime tends to contract, leaving with resulting from the phase where it is the seat of nuclear reactions a more or less compact residue. This phase of contraction increases considerably the magnetic field on the surface of the compact star. Thus, a white Naine has a magnetic field which can go up to 10 4 Teslas, whereas a neutron star young person, much more compact than dwarf white has a measured field with 10 8 even 10 9 Teslas. Certain neutron stars called abnormal pulsars X and Magnétar S seem to be equipped with a magnetic field up to 100 times more raised.
A magnet NdFeB (neodymium-iron-boron) of the size of a coin (creating a field of about 1,25 T:

$F= \ frac {\ driven g_1 g_2} {4 \ pi r^2}$
with F in newton, G and G the intensity of these poles in has · m, μ the magnetic permeability of the medium in T ·m·With -1 and R the distance enters the poles in meters.

Biological effects

Effect of the magnetostatic fields

The various known species are not identically sensitive to the electromagnetic fields. The data concerning the human beings are still sporadic. The static fields lower than 8 Tesla probably do not have notable physiological effects, if it is not the appearance at certain people of Phosphène S when they are exposed to fields of more than 4 T. the the World Health Organization undertakes still today studies on the possible hazards.
Continuous-current fields also intense are relatively difficult to obtain apart from the specialized laboratories, the current applications generally implying fields lower than the Tesla.
Current research is directed more towards the not-ionizing fields of very low frequency (EMF: extremely low frequency ), which is not static, but seems to act on the biological systems or sometimes to cause Cancer S.,

Effect of the pulsated magnetic fields

The pulsated fields, that one can create much more intense, cause moreover by induction an electromagnetic radiation. This one can interact with the biological systems, and its effect depends on the Radiorésistance of the exposed species. In particular, according to the frequency, such fields can cause ionizing radiations: Ultraviolet S, x-rays or gamma. Those are dangerous for health, and cause in particular the burn of fabrics.
Recently, of the alternative medicines utilizing pulsated weak magnetic fields claim to limit the Cancer S or the Multiple sclerosis. So of such fields seem dangerous, no serious scientific study does not support these allegations to date., On the other hand, the pulsated magnetic fields can influence balance and seem to decrease the symptoms of the bipolar Trouble.
The effects, mainly related to induction in the nerves, thus allow via the magnetic Stimulation transcranienne, the neurological Diagnostic of Pathologie S .

Geological effects

Certain rocks are rich in ferromagnetic materials , which are sensitive to the magnetic field. In particular, they lose their magnetic properties beyond some Température, known as Température of Curie.
The basaltic rocks resulting for example from the volcanos or the oceanic Rift S, are heated beyond this temperature in the Magma. When they cool, they regain their magnetic properties, and solidifies the orientation of the Terrestrial magnetic field. One observes this effect through the magnetic anomalies of the rocks. It is by the analysis of these rocks that one observed the inversions of the terrestrial field. ,
There exist also rocks, like the Hématite, whose magnetic properties are such as one observes the variations of field during their formation. The study of these rocks is also a crucial factor which supports the Plate tectonics .

Magnetic energy

See also: electromagnetic Energy

The presence of a magnetic field is expressed overall by an energy, known as “magnetic energy”. It is expressed by:

$\ mathcal E_B = \ iiint \ frac {B^2} {2 \ driven}$
with B the value of the magnetic field and μ the magnetic permeability in any point. The magnetic field having an infinite range, it would be necessary to carry out this integral on all space. In practice, a voluminal energy is defined:

$\ mathfrak e_B = \ frac {B^2} {2 \ driven}$

Calculation of the field

See also: Partial derivative equation

The calculation of the magnetic field creates by a system requires to solve rather complex differential equations. There exists for that a numerical multitude of methods as the Finite element method, the Méthode of the finished differences and the Méthode of finished volumes to quote only the most widespread methods. However, it is possible analytically to calculate the magnetic field in certain simple cases. Except contrary mention, the expressions given for the calculation of the magnetic field are expressed in the units IF. That explains in particular the factor $\ frac {1} {4 \ pi}$ .

Theorem of Amp

See also: Theorem of Amp

Starting from the observations revealing a bond between electric currents and magnetic field, Andre-Marie Ampère stated an initially phenomenologic law, which described the effect observed. Shown since, within the more general framework of the electromagnetism, this relation became the Théorème of Amp. It is not valid, in any rigor, that in the cases Magnétostatique S.
The original formulation of this theorem is the following one:

$\ oint_C \ mathbf B. \ mathrm D \ ell = \ mu_0 I_ {enl}$
with B the magnetic field, C a closed and directed curve and I_enl the “intertwined” intensity which crosses the surface delimited by C .
It can be written locally, one has then:

$\ nabla \ times \ mathbf B = \ mu_0 \ mathbf j$
where μ₀ is the magnetic permeability of the vacuum, and J the vector density of current.
Put at fault in the case of magnetic fields or electric depending on time, Maxwell introduced into 1861 the “displacement currents”, whose variation corrected this relation: it is the local equation of Maxwell-Amp. One can locally write it in the form:

$\ nabla \ times \ mathbf B = \ mu_0 \ mathbf J + \ mu_0 \ epsilon_0 \ frac {\ partial \ mathbf E} {\ partial T}$
with E the Electric field and ε₀ the electric permeability of the vacuum.
One can a posteriori rewrite this law in integral form, also called theorem of Amp:

$\ oint_C \ mathbf B. \ mathrm D \ ell = \ epsilon_0 (I_ {enl} + I_ {D})$
with

$I_ {D} = \ iint_S \ frac {\ partial \ mathbf E} {\ partial T}. D {\ mathbf S}$
where S is the surface delimited by contour C .
This is included/understood easily thanks to the theorem of Green-Stokes: $\ iint_S \ nabla \ times \ mathbf B. D {\ mathbf S} = \ oint_C \ mathbf B. \ mathrm D \ ell$ .

Local law of Biot-Savart

See also: Law of Biot-Savart

The Loi of Biot-Savart makes it possible to give the expression of the magnetic field in a medium of magnetic Perméabilité isotropic and homogeneous.
The field B generated in a point of coordinates R by a load Q moving, located in a point R' and moving at the speed v , is given by the following relation:

$\ mathbf {B} \ left ({\ mathbf {R}} \ right) = \ frac {\ driven} {4 \ pi}. \ frac {Q \ mathbf {v} \ times ({\ mathbf {R}} - {\ mathbf {r'}})}$

Integral law of Biot-Savart
If one deals with distribution of currents, which is known in any point, then one can integrate the local relation. With the preceding notations, that gives:

$\ mathbf B ({\ mathbf {R}}) = \ frac {\ driven} {4 \ pi} \ int \ frac {\ mathbf J \ left ({\ mathbf {r'}} \ right) \ times ({\ mathbf {R}} - {\ mathbf {r'}})}$

Potential vector
See also: Potential vector The magnetic absence of monopolies implies that the divergence of the magnetic field is null:

$\mathrm{div} \; \ mathbf B = 0$ .

This implies, according to the theorems of the vectorial Analyze, which there exists a vector field has , equal to the Rotationnel of B :

${\ mathbf {B}} = {\ mathrm belch} \; {\ mathbf {has}}$ .

Such a field has is called Potentiel vector, in opposition to the electric Potentiel, called “potential scalar”, of the Electric field.
This potential is not however single: it is defined except for a Gradient. Indeed, the rotational one of a gradient is identically null, also the potential vector has' defined by:

${\ mathbf {has}} '= {\ mathbf {has}} + \ nabla \ phi$
also check it the relation:

${\ mathbf {B}} = {\ mathrm belch} \; {\ mathbf {has}} '$ .

In a somewhat strange way, the fundamental quantity is not the magnetic field but the Potentiel vector, whereas this last cannot be defined in a univocal way. Such a situation is called in physics Invariance of gauge: identical phenomena, here the field B , can be generated by several configurations, or several “gauges” of the fundamental object, here the field has . From a mathematical point of view, the invariance of gauge is the cause of a fundamental law of electromagnetism, the Conservation of the electric charge. This law, checked in experiments with a very high degree of accuracy implies indeed that the fundamental object appearing in electromagnetism is neither the magnetic field nor the electric field, but the potential vector and electric potential.
Knowing has , one can easily deduce B from it. The fact that the potential vector evening more fundamental than the magnetic field shows through in quantum Mécanique, where in the presence of magnetic field, it is in fact the potential vector which appears in the equation of Schrödinger, which describes the evolution of the elementary particles. The most manifest illustration of the preeminence of the potential vector is in the Effet Aharonov-Bohm, where one is brought to consider configurations in which the field B is cancelled in certain areas whereas the potential vector has is not null and influences the behavior of the particles explicitly.
It is possible besides to calculate the potential vector has directly starting from the data of the currents:

$\ mathbf has \ left ({\ mathbf {R}} \ right) = \ frac {\ driven} {4 \ pi} \ int \ frac {\ mathbf J \ left ({\ mathbf {r'}} \ right)} {\ mathrm D} {\ mathbf {r'}}$ ,
the expression above being valid in any rigor only when the currents - thus fields - do not depend on the Temps. In practice, one neglects these variations as long as does not study the waves and their propagation.
In these last cases, it is necessary to replace the expression above by a more complex expression, calling upon the concept of potential delayed to take account of the travel time of the magnetic field.

Mathematical properties

Symmetries

In time that field pseudovectoriel, the magnetic field Indeed has a particular behavior compared to the Symétrie S., contrary to the field (vectorial) electric, the magnetic fields do not follow the symmetry of their sources. One speaks thus about “axial” vector or “Pseudovecteur”.
For example, for a circular whorl traversed by a current:

a symmetry plane Π + is that which contains the whorl;

a plan of antisymetry Π - is very plane passing by the center of the whorl and orthogonal in the foreground.

Respectively, Π + and Π - are a plan of antisymetry and symmetry for the magnetic field.

Change of reference frame

In traditional mechanics, where one considers relative speeds much lower than the Speed of light, the measured magnetic field is identical in two frames of reference in rectilinear translation and uniform one compared to the other (referential galiléens). This property is not shared by the Electric field, whose value changes from one reference frame to another if the magnetic field is nonnull.

Uses

Deviation of particles

One can show that a magnetic field affects the displacement of charged particles, by inflecting their trajectory, but without modifying the value their speed. It is thus used to curve their trajectory in the particle accelerators.
Indeed, according to the law of Lorentz, the force which a magnetic field B exerts on a particle of load Q moving with the Speed v is:

$\ mathbf {F} = Q \ mathbf {v} \ times \ mathbf {B}$
Thus, this force is always orthogonal at the speed, therefore its work is null:

$W = \ mathbf {F}. \ mathrm D \ mathbf R = 0$

Consequently, the standard speed is not influenced by the magnetic field. On the other hand, this force can modify the direction of this one.

Bubble chambers

See also: Bubble chamber

The magnetic field deviates the particles charged. If, moreover, the medium has a certain viscosity, then these particles describe spirals, which one can deduce the electric Charge (the direction of rolling up) and the Masse (through deceleration) from the particles.
It is the principle of the bubble chambers, invented at the beginning of the 20th century to observe, in particular, the components of the matter (Proton S, Neutron S and electron S), the Positon S and the Neutrino S. One prefers however today, since their invention in the years 1970, to use the rooms with wire.
In practice, there exists always a Electric field, which deviates the particles.
A particle in a bubble chamber is ideally subjected only to the magnetic force and the forces of friction. It thus checks:

$m \ frac {\ mathrm D \ mathbf v} {\ mathrm dt} = Q \ mathbf v \ times \ mathbf B - \ eta \ mathbf v$ .
what one can still write:

$\ dowry {\ mathbf v} - \ frac {Q} {m} \ mathbf v \ times \ mathbf B + \ frac {\ eta} {m} \ mathbf v = 0$ .

Magnetic resonance: IRM and NMR

See also: Imagery by magnetic resonance, nuclear Magnetic resonance

Magnetic resonance is a phenomenon which appears when certain atoms is placed in a magnetic field and receives an adapted radio operator radiation.
Indeed, the Atome S whose core is composed of an odd number of components - in particular the Hydrogène, of which the core summarizes itself with a Proton - present a kind of magnetic Moment, called magnetic Moment of spin. When a core is placed in a magnetic field - quantum mechanics obliges - it can be placed only in two distinct states. One can however make pass a core of one state to the other with a Photon of adapted pulsation: one speaks about Résonance. This phenomenon affecting the core of an atom, one speaks about nuclear Magnetic resonance.
An affected core turns over to balance by taking again its country of origin and by emitting a Photon. This radiation, in addition to indicating the presence of the core, can also inform on its vicinity within a Molécule. Indeed, it occurs couplings, which influence in particular its frequency. In NMR, one calls these variations with a solvent of reference “displacements”.
The Imagerie by nuclear magnetic resonance (IRM) is the application of this effect in Medical imagery, making it possible to have a sight 2D or 3D of part of the body, in particular of the Cerveau.

Electric transformers

See also: Electric transformer

A Electric transformer is a converter, which makes it possible to modify the values of the tension and the intensity of the current delivered by an alternative electric energy source in a system of tension and current of different values, but of the same Fréquence and of the same form. It carries out this transformation with an excellent output. It is similar to gears in mechanics (the couple on each toothed wheel being the analog of the tension and number of revolutions being the analog of the current).
A transformer consists of two parts: the Magnetic circuit and rollings up. Rollings up create or are crossed by a Magnetic flux that the magnetic circuit makes it possible to channel in order to limit the losses. In the case of a transformer perfect Single-phase current for which all the losses and the escapes of flow are neglected, the report/ratio of the number of whorl S primary educations and secondaries completely determines the report/ratio of transformation of the transformer. Thus, if one notes respectively note $n_1 \,$ and $n_2 \,$ the number of whorls to the primary education and the secondary, one obtains:

$\ frac {U_2} {U_1} = \ frac {n_2} {n_1}$

With $U_1 \,$ the primary tension and $U_2 \,$ the secondary tension.

Electric motors

See also: Wheel of Barlow, electric Machine

A electric machine is a device allowing the conversion of electrical energy into work or energy Mécanique: the rotary engines produce of a Couple by an angular displacement while the linear motors produce of a force by a linear displacement.
The forces generated by the magnetic fields, formulated by the relation of Lorentz, make it possible to consider devices which use such a field to transform the electromagnetic energy into mechanical energy.
The first electrical motor was built by Peter Barlow: a wheel, subjected to a permanent magnetic field, is traversed by a Electric current. He is thus exerted a force on this wheel, which is put then in rotation: it is the Roue of Barlow. It constitutes in fact the first electrical motor with D.C. current.
The bonds between magnetic field and electric field, expressed by the Maxwell's equations, make that it is possible to build systems which create a nonpermanent magnetic field — starting from a power source, by means of electromagnet S.
Within such apparatuses, one creates a revolving magnetic field, i.e. a field whose direction varies while turning in a direction or the other with a given rotational frequency.
One of the possibilities is to create such a field using fixed electromagnets - they constitute the “ stator ” - traversed by a variable Electric current of intensity, for example Triphasé. At the center, a part moving and sensitive to the magnetic field, made up for example of permanent magnets, is thus put moving: it is the “ rotor ”, whose rotation movement is transmitted to a tree. This principle for example is put work for the synchronous machines and the asynchronous machines
Another possibility is to create a permanent field with the stator using permanent magnets or of rollings up traversed by a D.C. current and to carry out a magnetic field turning to the rotor by a system of slipping connections so that this rotor field remains in squaring with the stator field. It is the principle put work for the Machine at D.C. current

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