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Magnetic moment
In physics, astronomy, chemistry, and electrical engineering, the term magnetic moment of a system (such as a loop of electric current, a bar magnet, an electron, a molecule, or a planet) usually refers to its magnetic dipole moment, and is a measure of the strength of the system's net magnetic source. Specifically, magnetic dipole moment quantifies the contribution of the system's internal magnetism to the external dipolar magnetic field produced by the system (i.e. the component of the external magnetic field that drops off with distance as the inverse square). Any dipolar magnetic field pattern is symmetric with respect to rotations around a particular axis, therefore it is customary to describe the magnetic dipole moment that creates such a field as a vector with a direction along that axis. For quadrupolar, octupolar, and higherorder multipole magnetic moments, see Multipole expansion. Additional recommended knowledge
Two kinds of magnetic sourcesFundamentally, contributions to any system's magnetic moment may come from sources of two kinds: (1) motion of electric charges, such as electric currents and (2) the intrinsic magnetism of elementary particles, such as the electron. 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. On the other hand, the magnitude of each elementary particle's intrinsic magnetic moment is 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. The direction of the magnetic moment of any elementary particle is entirely determined by the direction of its spin (the minus in front of the value above indicates that any electron's magnetic moment is antiparallel to its spin). Finally, the net magnetic moment of any system is a vector sum of contributions from one or both types of sources. For example, a hydrogen atom's magnetic moment is a vector sum of the following contributions: the intrinsic moment of the electron, the orbital motion of the electron around the proton, and the intrinsic moment of the proton. Formulas and values for calculating magnetic momentsIn the simplest case of a planar loop of electric current, its magnetic moment is defined as: where
By convention, the direction of the vector area is given by the right hand rule (moving one's right hand in the direction of the current around the loop, when the palm of the hand is "touching" the loop's outer edge, and the straight thumb indicates the direction of the vector area and thus of the magnetic moment). In case of an arbitrary closed loop of constant current I, the magnetic moment is given by where is the element of the vector area of the current loop. In the most general case of an arbitrary current distribution in space, the magnetic moment of such a distribution can be found from the following equation: where
The above equation can be used for calculating a magnetic moment of any assembly of moving charges, such as a spinning charged solid, by substituting
For example, the magnetic moment produced by an electric charge moving along a circular path is
where is the position of the charge q relative to the center of the circle and is the instantaneous velocity of the charge. For a free point charge moving in an external magnetic field the magnetic moment is a measure of the magnetic flux set up by the gyration of the charge in the magnetic field. The moment is opposite to the direction of magnetic field (i.e. it is diamagnetic) and is equal to the kinetic energy of the rotary motion divided by the magnetic field. For a spinning charged solid with a uniform charge density to mass density ratio, the ratio of its magnetic moment to its angular momentum, also known as gyromagnetic ratio, is equal to half the chargetomass ratio. This implies that a more massive assembly of charges spinning with the same angular momentum will have a proportionately weaker magnetic moment, compared to its lighter counterpart. Even though atomic particles cannot be accurately described as spinning charge distributions of uniform chargetomass ratio, this general trend can be sometimes observed in the atomic world, where intrinsic angular momenta of most particles are fairly constant: a small halfinteger (spin) times the reduced Planck constant . This is the basis for defining the magnetic moment units of Bohr magneton (assuming chargetomass ratio of the electron) and nuclear magneton (assuming chargetomass ratio of the proton). In atomic and nuclear physics, the symbol μ represents the magnitude of the magnetic moment, often measured in Bohr magnetons or nuclear 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:
For relation between the notions of magnetic moment and magnetization see magnetization. Magnetic field produced by a magnetic momentAny system possessing a net magnetic dipole moment will produce a dipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higherorder multipole components, those will drop off with distance more rapidly, so that only the dipolar component will dominate the magnetic field of the system at distances far away from it. Chosing a frame of reference in which the system center is at the origin, and the z axis is pointing in the direction of the system's magnetic moment simplifies the description of the magnetic field. In such frame of reference, the components of the dipolar magnetic field produced by the system, at any point (x,y,z) in space, are (in teslas):
Effects of an external magnetic field on a magnetic momentEquivalently, the magnetic moment of an object can be defined as a vector relating the aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by where
A magnetic moment in an externallyproduced magnetic field has a potential energy U: In a case when the external magnetic field is nonuniform, there will be a force, proportional to the magnetic field gradient, acting on the magnetic moment itself. Magnetic poles, analogy with the electric dipole momentMagnetic moment can be visualized as a bar magnet which has magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic 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: on both the strength p of its poles, and on the distance d separating them. The force is proportional to the product , where describes the "magnetic moment" or "dipole moment" of the magnet along a distance R and its direction as the angle between R and the axis of the bar magnet. These equations are completely analogous to the case of electric dipole moment. Magnetic moment of electronsElectrons and many nuclei also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsic angular momentum of the particles as discussed in the article electron magnetic dipole moment. It is these intrinsic magnetic moments that give rise to the macroscopic effects of magnetism, and other phenomena, such as nuclear magnetic resonance. The magnetic moment of the electron is where
and
Again it is important to notice that is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum. 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 spins 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 angular momentum. Magnetic moments of nuclei
The nuclear system is a complex physical system consisting of nucleons, i.e., protons and neutrons. The quantum mechanical properties of the nucleons include the spin among others. Since the electromagnetic moments of the nucleus depends 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. The nuclear magnetic moment 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 wavefunction. 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. Magnetic moments of moleculesAny molecule has a welldefined magnitude of magnetic moment, which may depend on the molecule's energy state. Typically, the overall magnetic moment of a molecule is a combination of the following contributions, in the order of their typical strength:
Examples of molecular magnetism
See alsoNotes


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Magnetic_moment". A list of authors is available in Wikipedia. 