Inelectromagnetism, themagnetic moment ormagnetic dipole moment is a vector quantity which characterizes the strength and orientation of amagnet or other object or system that exerts amagnetic field. The magnetic dipole moment of an object determines the magnitude oftorque the object experiences in a given magnetic field. When the same magnetic field is applied, objects with larger magnetic moments experience larger torques. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field. Its direction points from the south pole to the north pole of the magnet (i.e., inside the magnet).
The magnetic moment also expresses the magnetic force effect of a magnet. The magnetic field of a magnetic dipole is proportional to its magnetic dipole moment. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object.
The magnetic moment can be defined as a vector (reallypseudovector) relating the aligningtorque on the object from an externally appliedmagnetic field to the field vector itself. The relationship is given by:[1]whereτ is the torque acting on the dipole,B is the external magnetic field, andm is the magnetic moment.
This definition is based on how one could, in principle, measure the magnetic moment of an unknown sample. For a current loop, this definition leads to the magnitude of the magnetic dipole moment equaling the product of the current times the area of the loop. Further, this definition allows the calculation of the expected magnetic moment for any known macroscopic current distribution.
An alternative definition is useful forthermodynamics calculations of the magnetic moment. In this definition, the magnetic dipole moment of a system is the negative gradient of its intrinsic energy,Uint, with respect to external magnetic field:
Generically, the intrinsic energy includes the self-field energy of the system plus the energy of the internal workings of the system. For example, for a hydrogen atom in a 2p state in an external field, the self-field energy is negligible, so the internal energy is essentially the eigenenergy of the 2p state, which includes Coulomb potential energy and the kinetic energy of the electron. The interaction-field energy between the internal dipoles and external fields is not part of this internal energy.[2]
The unit for magnetic moment inInternational System of Units (SI)base units is A⋅m2, where A isampere (SI base unit of current) and m ismeter (SI base unit of distance). This unit has equivalents in other SI derived units including:[3][4]where N isnewton (SI unit of force), T istesla (SI unit of magnetic flux density), and J isjoule (SI unit ofenergy).[5]: 20–21
In theCGS system, there are several different sets of electromagnetism units, of which the main ones areESU,Gaussian, andEMU. Among these, there are alternative (non-equivalent) units of magnetic dipole moment:where statA isstatampere, cm iscentimeter, erg iserg, and G isgauss. The ratio of these two non-equivalent CGS units (EMU/ESU) is equal to thespeed of light in free space, expressed incm⋅s−1.
All formulas in this article are correct inSI units; they may need to be changed for use in other unit systems. For example, in SI units, a loop of current with currentI and areaA has magnetic momentIA (see below), but inGaussian units the magnetic moment isIA/c.
The magnetic moments of objects are typically measured with devices calledmagnetometers, though not all magnetometers measure magnetic moment: Some are configured to measuremagnetic field instead. If the magnetic field surrounding an object is known well enough, though, then the magnetic moment can be calculated from that magnetic field.[citation needed]
The magnetic moment is a quantity that describes the magnetic strength of an entire object. Sometimes, though, it is useful or necessary to know how much of the net magnetic moment of the object is produced by a particular portion of that magnet. Therefore, it is useful to define the magnetizationM as:wheremΔV andVΔV are the magnetic dipole moment and volume of a sufficiently small portion of the magnetΔV. This equation is often represented using derivative notation such thatwheredm is the elementary magnetic moment anddV is thevolume element. The net magnetic moment of the magnetm therefore iswhere the triple integral denotes integration over the volume of themagnet. For uniform magnetization (where both the magnitude and the direction ofM is the same for the entire magnet (such as a straight bar magnet) the last equation simplifies to:whereV is the volume of the bar magnet.
The magnetization is often not listed as a material parameter for commercially availableferromagnetic materials, though. Instead the parameter that is listed isresidual flux density (or remanence), denotedBr. The formula needed in this case to calculatem in (units of A⋅m2) is:
where:
Br is the residual flux density, expressed inteslas.
V is the volume of the magnet (in m3).
μ0 is the permeability of vacuum (4π×10−7 H/m).[6]
The preferred classical explanation of a magnetic moment has changed over time. Before the 1930s, textbooks explained the moment using hypothetical magnetic point charges. Since then, most have defined it in terms of Ampèrian currents.[7] In magnetic materials, the cause of the magnetic moment are thespin and orbital angular momentum states of theelectrons, and varies depending on whether atoms in one region are aligned with atoms in another.[citation needed]
An electrostatic analog for a magnetic moment: two opposing charges separated by a finite distance.
The sources of magnetic moments in materials can be represented by poles in analogy toelectrostatics. This is sometimes known as the Gilbert model.[8]: 258 In this model, a small magnet is modeled by a pair offictitious magnetic monopoles of equal magnitude but oppositepolarity. Each pole is the source of magnetic force which weakens with distance. Sincemagnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strengthp of its poles (magnetic pole strength), and the vector separating them. The magnetic dipole momentm is related to the fictitious poles as[7]
It points in the direction from the South to the North pole. The analogy with electric dipoles should not be taken too far because magnetic dipoles are associated withangular momentum (seeRelation to angular momentum). Nevertheless, magnetic poles are very useful formagnetostatic calculations, particularly in applications toferromagnets.[7] Practitioners using the magnetic pole approach generally represent themagnetic field by theirrotational fieldH, in analogy to theelectric fieldE.
The Amperian loop model: A current loop (ring) that goes into the page at the x and comes out at the dot produces aB-field (lines). The north pole is to the right and the south to the left.
AfterHans Christian Ørsted discovered that electric currents produce a magnetic field andAndré-Marie Ampère discovered thatelectric currents attract and repel each other similar to magnets, it was natural to hypothesize that all magnetic fields are due to electric current loops. In this model developed by Ampère, the elementary magnetic dipole that makes up all magnets is a sufficiently small amperian loop of currentI. The dipole moment of this loop iswhereS is the area of the loop. The direction of the magnetic moment is in a direction normal to the area enclosed by the current consistent with the direction of the current using the right hand rule.
Magnetic moment of a planar current density having magnitudeI and enclosing an areaS or, equivalently, a loop of currentI enclosingS
The magnetic dipole moment can be calculated for a localized (does not extend to infinity) current distribution assuming that we know all of the currents involved. Conventionally, the derivation starts from amultipole expansion of thevector potential. This leads to the definition of the magnetic dipole moment as:where× is thevector cross product,r is the position vector, andj is theelectric current density and the integral is a volume integral.[9]: § 5.6 When the current density in the integral is replaced by a loop of current I in a plane enclosing an area S then thevolume integral becomes aline integral and the resulting dipole moment becomeswhich is how the magnetic dipole moment for an Amperian loop is derived.
Practitioners using the current loop model generally represent the magnetic field by thesolenoidal fieldB, analogous to the electrostatic fieldD.
A generalization of the above current loop is a coil, orsolenoid. Its moment is the vector sum of the moments of individual turns. If the solenoid hasN identical turns (single-layer winding) and vector areaS,
When calculating the magnetic moments of materials or molecules on the microscopic level it is often convenient to use a third model for the magnetic moment that exploits the linear relationship between theangular momentum and the magnetic moment of a particle. While this relation is straightforward to develop for macroscopic currents using the amperian loop model (seebelow), neither the magnetic pole model nor the amperian loop model truly represents what is occurring at the atomic and molecular levels. At that levelquantum mechanics must be used. Fortunately, the linear relationship between the magnetic dipole moment of a particle and its angular momentum still holds, although it is different for each particle. Further, care must be used to distinguish between the intrinsic angular momentum (orspin) of the particle and the particle's orbital angular momentum. Seebelow for more details.
The torqueτ on an object having a magnetic dipole momentm in a uniform magnetic fieldB is:
This is valid for the moment due to any localized current distribution provided that the magnetic field is uniform. For non-uniform B the equation is also valid for the torque about the center of the magnetic dipole provided that the magnetic dipole is small enough.[8]: 257
An electron, nucleus, or atom placed in a uniform magnetic field will precess with a frequency known as theLarmor frequency. SeeResonance.
A magnetic moment in an externally produced magnetic field has a potential energyU:
In a case when the external magnetic field is non-uniform, there will be a force, proportional to the magnetic fieldgradient, acting on the magnetic moment itself. There are two expressions for the force acting on a magnetic dipole, depending on whether themodel used for the dipole is a current loop or two monopoles (analogous to the electric dipole).[10] The force obtained in the case of a current loop model is
Assuming existence of magnetic monopole, the force is modified as follows:
In the case of a pair of monopoles being used (i.e. electric dipole model), the force isAnd one can be put in terms of the other via the relation
In all these expressionsm is the dipole andB is the magnetic field at its position. If there are no currents or time-varying electrical fields or magnetic charge,∇×B = 0,∇⋅B = 0 and the two expressions agree.
One can relate the magnetic moment of a system to the free energy of that system.[11] In a uniform magnetic fieldB, thefree energyF can be related to the magnetic momentM of the system aswhereS is theentropy of the system andT is the temperature. Therefore, the magnetic moment can also be defined in terms of the free energy of a system as
In addition, an applied magnetic field can change the magnetic moment of the object itself; for example by magnetizing it. This phenomenon is known asmagnetism. An applied magnetic field can flip the magnetic dipoles that make up the material causing bothparamagnetism andferromagnetism. Additionally, the magnetic field can affect the currents that create the magnetic fields (such as the atomic orbits) which causesdiamagnetism.
Magnetic field lines around a "magnetostatic dipole". The magnetic dipole itself is located in the center of the figure, seen from the side, and pointing upward.
Any system possessing a net magnetic dipole momentm will produce adipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higher-ordermultipole components, those will drop off with distance more rapidly, so that only the dipole component will dominate the magnetic field of the system at distances far away from it.
The magnetic field of a magnetic dipole depends on the strength and direction of a magnet's magnetic moment but drops off as the cube of the distance such that:
where is themagnetic field produced by the magnet and is a vector from the center of the magnetic dipole to the location where the magnetic field is measured. The inverse cube nature of this equation is more readily seen by expressing the location vector as the product of its magnitude times the unit vector in its direction () so that:
The equivalent equations for the magnetic-field are the same except for a multiplicative factor ofμ0 =4π×10−7H/m, whereμ0 is known as thevacuum permeability. For example:
As discussed earlier, the force exerted by a dipole loop with momentm1 on another with momentm2 iswhereB1 is the magnetic field due to momentm1. The result of calculating the gradient is[12][13]wherer̂ is the unit vector pointing from magnet 1 to magnet 2 andr is the distance. An equivalent expression is[13]The force acting onm1 is in the opposite direction.
The magnetic field of any magnet can be modeled by a series of terms for which each term is more complicated (having finer angular detail) than the one before it. The first three terms of that series are called themonopole (represented by an isolated magnetic north or south pole) thedipole (represented by two equal and opposite magnetic poles), and thequadrupole (represented by four poles that together form two equal and opposite dipoles). The magnitude of the magnetic field for each term decreases progressively faster with distance than the previous term, so that at large enough distances the first non-zero term will dominate.[citation needed]
For many magnets the first non-zero term is the magnetic dipole moment. (To date, no isolatedmagnetic monopoles have been experimentally detected.) A magnetic dipole is the limit of either a current loop or a pair of poles as the dimensions of the source are reduced to zero while keeping the moment constant. As long as these limits only apply to fields far from the sources, they are equivalent. However, the two models give different predictions for the internal field (see below).
Traditionally, the equations for the magnetic dipole moment (and higher order terms) are derived from theoretical quantities calledmagnetic potentials[9]: § 5.6 which are simpler to deal with mathematically than the magnetic fields.[citation needed]
In the magnetic pole model, the relevant magnetic field is thedemagnetizing field. Since the demagnetizing portion of does not include, by definition, the part of due to free currents, there exists amagnetic scalar potential such that
In the amperian loop model, the relevant magnetic field is the magnetic induction. Since magnetic monopoles do not exist, there exists amagnetic vector potential such that
Both of these potentials can be calculated for any arbitrary current distribution (for the amperian loop model) or magnetic charge distribution (for the magnetic charge model) provided that these are limited to a small enough region to give:where is thecurrent density in the amperian loop model, is the magnetic pole strength density in analogy to the electriccharge density that leads to the electric potential, and the integrals are the volume (triple) integrals over the coordinates that make up. The denominators of these equation can be expanded using themultipole expansion to give a series of terms that have larger of power of distances in the denominator. The first nonzero term, therefore, will dominate for large distances. The first non-zero term for the vector potential is:where is:where× is thevector cross product,r is the position vector, andj is theelectric current density and the integral is a volume integral.
In the magnetic pole perspective, the first non-zero term of thescalar potential is
Here may be represented in terms of the magnetic pole strength density but is more usefully expressed in terms of themagnetization field as:
The same symbol is used for both equations since they produce equivalent results outside of the magnet.
External magnetic field produced by a magnetic dipole moment
The two models for a dipole (magnetic poles or current loop) give the same predictions for the magnetic field far from the source. However, inside the source region, they give different predictions. The magnetic field between poles (see the figure forMagnetic pole model) is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). The limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.[7]
If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole charge and distance constant, the limiting field is[7]
The fieldsH andB are related by whereM(r) =mδ(r) is themagnetization.
If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field isUnlike the expressions in the previous section, this limit is correct for the internal field of the dipole.[7][9]: 184
Viewing a magnetic dipole as current loop brings out the close connection between magnetic moment and angular momentum. Since the particles creating the current (by rotating around the loop) have charge and mass, both the magnetic moment and the angular momentum increase with the rate of rotation. The ratio of the two is called thegyromagnetic ratio or so that:[14][15]where is the angular momentum of the particle or particles that are creating the magnetic moment.
In the amperian loop model, which applies for macroscopic currents, the gyromagnetic ratio is one half of thecharge-to-mass ratio. This can be shown as follows. The angular momentum of a moving charged particle is defined as:whereμ is the mass of the particle andv is the particle'svelocity. The angular momentum of the very large number of charged particles that make up a current therefore is:whereρ is themass density of the moving particles. By convention the direction of the cross product is given by theright-hand rule.[16]
This is similar to the magnetic moment created by the very large number of charged particles that make up that current:where and is thecharge density of the moving charged particles.
Comparing the two equations results in:where is the charge of the particle and is the mass of the particle.
Even though atomic particles cannot be accurately described as orbiting (and spinning) charge distributions of uniform charge-to-mass ratio, this general trend can be observed in the atomic world so that:where theg-factor depends on the particle and configuration. For example, theg-factor for the magnetic moment due to an electron orbiting a nucleus is one while theg-factor for the magnetic moment of electron due to its intrinsic angular momentum (spin) is a little larger than 2. Theg-factor of atoms and molecules must account for the orbital and intrinsic moments of its electrons and possibly the intrinsic moment of its nuclei as well.
Contributions due to the sources of the first kind can be calculated from knowing the distribution of all the electric currents (or, alternatively, of all the electric charges and their velocities) inside the system, by using the formulas below.
Contributions due to particle spin sum themagnitude of each elementary particle's intrinsic magnetic moment, a fixed number, often measured experimentally to a great precision. For example, any electron's magnetic moment is measured to be−9.284764×10−24 J/T.[17] Thedirection of the magnetic moment of any elementary particle is entirely determined by the direction of itsspin, with thenegative value indicating that any electron's magnetic moment is antiparallel to its spin.
The net magnetic moment of any system is avector sum of contributions from one or both types of sources.For example, the magnetic moment of an atom ofhydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an electron) is a vector sum of the following contributions:
the intrinsic moment of the electron,
the orbital motion of the electron around the proton,
the intrinsic moment of the proton.
Similarly, the magnetic moment of abar magnet is the sum of the contributing magnetic moments, which include the intrinsic and orbital magnetic moments of the unpairedelectrons of the magnet's material and the nuclear magnetic moments.
For an atom, individual electron spins are added to get a total spin, and individual orbital angular momenta are added to get a total orbital angular momentum. These two then are added usingangular momentum coupling to get a total angular momentum. For an atom with no nuclear magnetic moment, the magnitude of the atomic dipole moment,, is then[18]wherej is thetotal angular momentum quantum number,gJ is theLandég-factor, andμB is theBohr magneton. The component of this magnetic moment along the direction of the magnetic field is then[19]
The negative sign occurs because electrons have negative charge.
Theintegerm (not to be confused with the moment,) is called themagnetic quantum number or theequatorial quantum number, which can take on any of2j + 1 values:[20]
Due to the angular momentum, the dynamics of a magnetic dipole in a magnetic field differs from that of an electric dipole in an electric field. The field does exert a torque on the magnetic dipole tending to align it with the field. However, torque is proportional to rate of change of angular momentum, soprecession occurs: the direction of spin changes. This behavior is described by theLandau–Lifshitz–Gilbert equation:[21][22]whereγ is thegyromagnetic ratio,m is the magnetic moment,λ is the damping coefficient andHeff is the effective magnetic field (the external field plus any self-induced field). The first term describes precession of the moment about the effective field, while the second is a damping term related to dissipation of energy caused by interaction with the surroundings.
Electrons and many elementary particles also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsicangular momentum of the particles as discussed in the articleElectron magnetic moment. It is these intrinsic magnetic moments that give rise to the macroscopic effects ofmagnetism, and other phenomena, such aselectron paramagnetic resonance.[citation needed]
Again it is important to notice thatm is a negative constant multiplied by thespin, so the magnetic moment of the electron is antiparallel to the spin. This can be understood with the following classical picture: if we imagine that the spin angular momentum is created by the electron mass spinning around some axis, the electric current that this rotation creates circulates in the opposite direction, because of the negative charge of the electron; such current loops produce a magnetic moment which is antiparallel to the spin. Hence, for a positron (the anti-particle of the electron) the magnetic moment is parallel to its spin.
The nuclear system is a complex physical system consisting of nucleons, i.e.,protons andneutrons. The quantum mechanical properties of the nucleons include the spin among others. Since the electromagnetic moments of the nucleus depend on the spin of the individual nucleons, one can look at these properties with measurements of nuclear moments, and more specifically the nuclear magnetic dipole moment.
Most common nuclei exist in theirground state, although nuclei of someisotopes have long-livedexcited states. Eachenergy state of a nucleus of a given isotope is characterized by a well-defined magnetic dipole moment, the magnitude of which is a fixed number, often measured experimentally to a great precision. This number is very sensitive to the individual contributions from nucleons, and a measurement or prediction of its value can reveal important information about the content of the nuclear wave function. There are several theoretical models that predict the value of the magnetic dipole moment and a number of experimental techniques aiming to carry out measurements in nuclei along the nuclear chart.
Any molecule has a well-defined magnitude of magnetic moment, which may depend on the molecule'senergy state. Typically, the overall magnetic moment of a molecule is a combination of the following contributions, in the order of their typical strength:
Thedioxygen molecule, O2, exhibits strongparamagnetism, due to unpaired spins of its outermost two electrons.
Thecarbon dioxide molecule, CO2, mostly exhibitsdiamagnetism, a much weaker magnetic moment of the electronorbitals that is proportional to the external magnetic field. The nuclear magnetism of a magneticisotope such as13C or17O will contribute to the molecule's magnetic moment.
Thedihydrogen molecule, H2, in a weak (or zero) magnetic field exhibits nuclear magnetism, and can be in apara- or anortho- nuclear spin configuration.
Many transition metal complexes are magnetic. The spin-only formula is a good first approximation for high-spin complexes of first-rowtransition metals.[23]
In atomic and nuclear physics, the Greek symbolμ represents themagnitude of the magnetic moment, often measured in Bohr magnetons ornuclear magnetons, associated with the intrinsic spin of the particle and/or with the orbital motion of the particle in a system. Values of the intrinsic magnetic moments of some particles are given in the table below:
Intrinsic magnetic moments and spins of some elementary particles[24]
^Landau, L. D.; Lifshitz, E. M.; Pitaevskii, L. P. (January 15, 1984).Electrodynamics of Continuous Media: Volume 8 (Course of Theoretical Physics) (2 ed.). Butterworth-Heinemann. p. 130.ISBN978-0-7506-2634-7.
^Figgis, B.N.; Lewis, J. (1960)."The magnetochemistry of complex compounds". In Lewis, J.; Wilkins, R.G. (eds.).Modern Coordination Chemistry: Principles and methods. New York: Interscience. pp. 405–407.
^"Search results matching 'magnetic moment'".CODATA internationally recommended values of the Fundamental Physical Constants. National Institute of Standards and Technology. Retrieved11 May 2012.
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