User:老陳/sandbox2

The magnetic moment of a magnet is a measure of its tendency to align with a magnetic field. Both the magnetic moment and magnetic field may be considered to be vectors having a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet. The magnetic field produced by a magnet is proportional to its magnetic moment as well. For example, a loop of electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments. More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field. 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.

Units

In the International System of Units (SI), the dimension of magnetic dipole moment is Area×current, or L²I (see examples below). This is the basis on which the unit of magnetic dipole moment is defined. The SI unit of magnetic dipole moment has two equivalent representations:

1 m2·A = 1 J/T.

In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment in CGS:

(ESU CGS) 1 statA·cm² = 3.33564095×10^-14 (m²·A or J/T)

and (more frequently used)

(EMU CGS and Gaussian-CGS) 1 erg/G = 1 abA·cm² = 10^-3 (m²·A or J/T).

The ratio of these two non-equivalent CGS units (EMU/ESU) is equal exactly to the speed of light in free space, expressed in cm/s.

All formulas in this article are correct in SI units, but in other unit systems, the formulas may need to be changed. For example, in SI units, a loop of current with current I and area A has magnetic moment I×A (see below), but in Gaussian units the magnetic moment is I×A/c.

Two kinds of magnetic sources

Fundamentally, 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).

The net magnetic moment of any system is a vector sum of contributions from one or both types of sources. For example, the magnetic moment of an atom of hydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an electron) is a vector sum of the following contributions: (1) the intrinsic moment of the electron, (2) the orbital motion of the electron around the proton, (3) the intrinsic moment of the proton. Similarly, the magnetic moment of a bar magnet is the sum of the intrinsic and orbital magnetic moments of the unpaired electrons of the magnet's material.

Magnetism and angular momentum

There exists a close connection between angular momentum and magnetism, expressed on a macroscopic scale in the Einstein-de Haas effect, or "rotation by magnetization," and its inverse, the Barnett effect, or "magnetization by rotation."

At the atomic and sub-atomic scales, this connection is expressed by the ratio of magnetic moment to angular momentum, the gyromagnetic ratio.

Examples of magnetic moments

Magnetic moment of circular coils

Magnetic moment of a current-carrying loop depends on its area and current. For example, the magnitude of magnetic moment for a single-turn circular coil of radius 5 cm carrying 1 A of current is determined as:

$\pi \times (0.05\ \mathrm {m} )^{2}\times (1\;\mathrm {A} )\approx 0.008\;\mathrm {A} \cdot \mathrm {m} ^{2}=0.008\;\mathrm {J/T} \;.$

The vector of this moment is pointing perpendicular to the plane of the loop, in the direction of the magnetic field at the center of the loop. (See the right-hand rule.) Knowing this value of the loop's magnetic moment can be used to establish the following facts:

• At distances R much larger than the radius of the loop r = 0.05 m, the magnetic field produced by this loop will drop off as:

$(10^{-7}\;\mathrm {T} \cdot \mathrm {m/A} )\times 2\times {\frac {0.008\;\mathrm {A} \cdot \mathrm {m} ^{2}}{R^{3}}}\approx {\frac {1.6\times 10^{-9}\;\mathrm {T} \cdot \mathrm {m} ^{3}}{R^{3}}}$ (along the loop's axis)

and

$-(10^{-7}\;\mathrm {T} \cdot \mathrm {m/A} )\times {\frac {0.008\;\mathrm {A} \cdot \mathrm {m} ^{2}}{R^{3}}}\approx -{\frac {0.8\times 10^{-9}\;\mathrm {T} \cdot \mathrm {m} ^{3}}{R^{3}}}$ (in the loop's plane). Minus indicates the field direction opposite to the axial case. $10^{-7}={\mu _{0}}/{4\pi }\,\!$, see magnetic field produced by a magnetic moment for more details.

• In the Earth's magnetic field of 0.5 G (5×10⁻⁵ T) perpendicular to the loop's axis, the loop (as well as the Earth) will experience a torque in newton-meters of

$\tau \approx (0.008\;\mathrm {J/T} )\times (5\times 10^{-5}\;\mathrm {T} )=4\times 10^{-7}\ \mathrm {N} \cdot \mathrm {m}$.

This torque can be used to make an electric compass.

• If such compass is allowed to orient its axis parallel to the Earth's field, the amount of energy in joules released from the compass-Earth system is given by:

$U\approx 0.008\;\mathrm {J/T} \times 5\times 10^{-5}\;\mathrm {T} =4\times 10^{-7}\ \mathrm {J}$.

This energy can be dissipated into heat to overcome friction in the compass suspension system.

Magnetic moment of solenoids

Magnetic moment of a multi-turn coil (or solenoid) is determined as the vector sum of the moments of individual turns. In the case of identical turns (single-layer winding), it is equal simply to the individual turn's moment multiplied by the total number of turns in the solenoid. Once the value of the total magnetic moment is found, it can be used to establish the far field, the torque, and the stored energy in the external field in the same way as for the single-turn loop.

Magnetic dipoles

See also: Spin magnetic moment and Micromagnetism

Magnetic field lines around a ”magnetostatic dipole” the magnetic dipole itself is in the center and is seen from the side.

The magnetic field of an ideal magnetic dipole is depicted on the right. As discussed below, however, due to the inherent connection between angular momentum and magnetism, magnetic dipoles in actual materials are not ideal magnetic dipoles. (As mentioned above, the connection between angular momentum and magnetism is the basis of the Einstein-de Haas effect "rotation by magnetization" and its inverse, the Barnett effect or "magnetization by rotation".)

The magnetic field of permanent magnets and of all magnetic material originate at the atomic level. The total magnetic moment of an atom is due to a combination of 'currents' of electrons 'orbiting' the nuclei of the magnetic material plus a spin component of the magnetic moment of the electrons and the nucleus. (The true nature of the internal magnetic field of the electrons and of the nucleons that make up the nucleus is relativistic in nature.) The orbital component of these tiny magnets can be modeled as tiny loops of current with associated magnetic dipoles. The dipole moment of that dipole is defined as the current times the area of the loop and represents the strength of that magnet (magnetic dipole). In magnetic materials; such as alloys of iron, cobalt and nickel; however, the magnetism is almost entirely spin magnetism, not orbital magnetism.

The magnetic dipole originating in an atom, electron, or nucleus is not a true dipole, as is an electric dipole. Viewing a magnetic dipole as a rotating charged sphere brings out the close connection between magnetic moment and angular momentum. Both the magnetic moment and the angular momentum increase with the rate of rotation of the sphere. The ratio of the two is called the gyromagnetic ratio, usually denoted by the symbol γ.

Magnetic moment of atoms

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 using angular momentum coupling to get a total angular momentum. The magnitude of the atomic dipole moment is then:

$m_{\mathrm {Atom} }=g_{J}{\mu }_{B}{\sqrt {J(J+1)}}\ ,$

where J is the total angular momentum quantum number, g_J is the Landé g-factor, and μ_B is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then:

$m_{\mathrm {Atom} }(z)=-mg_{J}{\mu }_{B}\ ,$

where m is called the magnetic quantum number or the equatorial quantum number, which can take on any of 2J+1 values: -J, −(J-1), … , (J−1), J. The negative sign occurs because electrons have negative charge.

Because of 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, so precession occurs: the direction of spin changes. This behavior is described by the Landau-Lifshitz-Gilbert equation:

${\frac {1}{\gamma }}{\frac {d\mathbf {m} }{dt}}=\mathbf {m\times H_{eff}} -{\frac {\lambda }{\gamma m}}\mathbf {m\times } {\frac {d\mathbf {m} }{dt}}\ ,$

where γ =gyromagnetic ratio, m = magnetic moment, λ = damping coefficient and H_eff = effective magnetic field (the external field plus any self-field), and '×' = vector cross product. 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 surroundings.

Magnetic moment of electrons

Electrons and many elementary particles 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 electron paramagnetic resonance.

The magnetic moment of the electron is

${\boldsymbol {\mu }}_{S}=-g_{S}\mu _{B}({\boldsymbol {S}}/\hbar )$

where

$\mu _{B}\,$ is the Bohr magneton, S is electron spin,

and the electron g-factor is

$g_{S}=2\;$ in Dirac mechanics, but is slightly larger,

$g_{S}=2.002\,319\,304\,36\;$ in reality due to quantum electrodynamics effects.

Again it is important to notice that ${\boldsymbol {\mu }}$ is a negative constant multiplied by the spin, so the magnetic moment 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, a positron (the anti-particle of the electron) with its positive charge has its magnetic moment parallel to its spin.

Magnetic moments of nuclei

See also: Nuclear magnetic moment

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 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 their ground state, although nuclei of some isotopes have long-lived excited states. Each energy 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.

Magnetic moments of molecules

Any molecule has a well-defined 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:

• magnetic moments due to its unpaired electron spins (paramagnetic contribution), if any
• orbital motion of its electrons, which in the ground state is often proportional to the external magnetic field (diamagnetic contribution)
• the combined magnetic moment of its nuclear spins, which depends on the nuclear spin configuration.

Examples of molecular magnetism

• Oxygen molecule, O₂, exhibits strong paramagnetism, due to unpaired spins of its outermost two electrons.
• Carbon dioxide molecule, CO₂, mostly exhibits diamagnetism, a much weaker magnetic moment of the electron orbitals that is proportional to the external magnetic field. In the rare instance when a magnetic isotope, such as ¹³C or ¹⁷O, is present, it will contribute its nuclear magnetism to the molecule's magnetic moment.
• Hydrogen molecule, H₂, in a weak (or zero) magnetic field exhibits nuclear magnetism, and can be in a para- or an ortho- nuclear spin configuration.

Formulas for calculating and values of magnetic moments

Planar loop

In the simplest case of a planar loop of electric current, its magnetic moment is defined as:

${\boldsymbol {\mu }}=I\mathbf {a}$

where

${\boldsymbol {\mu }}$ is the magnetic moment, a vector measured in ampere–square meters, or equivalently in joules per tesla,

$\mathbf {a}$ is the vector area of the current loop, measured in square meters (x, y, and z coordinates of this vector are the areas of projections of the loop onto the yz, zx, and xy planes), and

$~I$ is the current in the loop (assumed to be constant), a scalar measured in amperes.

By convention, the direction of the vector area is given by the right hand grip rule (curling the fingers of 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).

Arbitrary closed loop

In case of an arbitrary closed loop of constant current $I$, the magnetic moment is given by

${\boldsymbol {\mu }}=I\int d\mathbf {a}$

where $d\mathbf {a}$ is the element of the vector area of the current loop.

Arbitrary current distribution

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:

${\boldsymbol {\mu }}={\frac {1}{2}}\int \mathbf {r} \times \mathbf {J} \,dV$

where

$dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi$ is the volume element, $\mathbf {r}$ is the position vector pointing from the origin to the location of the volume element, and J is the current density vector at that location.

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

$\mathbf {J} =\rho \mathbf {v}$ where $\rho$ is the electric charge density at a given point and $\mathbf {v}$ is the instantaneous linear velocity of that point.

For example, the magnetic moment produced by an electric charge moving along a circular path is

${\boldsymbol {\mu }}={\frac {1}{2}}\,q\,\mathbf {r} \times \mathbf {v}$,

where $\mathbf {r}$ is the position of the charge $q$ relative to the center of the circle and $\mathbf {v}$ 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 charge-to-mass 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 charge-to-mass ratio, this general trend can be sometimes observed in the atomic world, where intrinsic angular momenta of most particles are fairly constant: a small half-integer (spin) times the reduced Planck constant $\hbar$. This is the basis for defining the magnetic moment units of Bohr magneton (assuming charge-to-mass ratio of the electron) and nuclear magneton (assuming charge-to-mass ratio of the proton).

Elementary particles

In atomic and nuclear physics, the symbol $\mu$ 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:

Intrinsic magnetic moments and spins of some elementary particles

Particle	Magnetic dipole moment in SI units (10−27 J/T)	Spin quantum number (dimensionless)
electron	-9284.764	1/2
proton	+14.106067	1/2
neutron	-9.66236	1/2
muon	-44.904478	1/2
deuteron	+4.3307346	1
triton	+15.046094	1/2

For relation between the notions of magnetic moment and magnetization see magnetization.

Magnetic field produced by a magnetic moment

Any system possessing a net magnetic dipole moment ${\boldsymbol {\mu }}$ 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 higher-order 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.

Choosing 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 ${\boldsymbol {\mu }}$ 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):

$B_{x}(x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,\,3\mu \,{\frac {xz}{(x^{2}+y^{2}+z^{2})^{5/2}}}$

$B_{y}(x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,\,3\mu \,{\frac {yz}{(x^{2}+y^{2}+z^{2})^{5/2}}}$

$B_{z}(x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,\,3\mu \,{\frac {\,z^{2}\!-{\frac {1}{3}}\,(x^{2}+y^{2}+z^{2})\,}{(x^{2}+y^{2}+z^{2})^{5/2}}}\,$,

as well as the 'transverse' component:

$B_{\perp }(x,y,z)\,=\,{\sqrt {B_{x}^{2}(x,y,z)+B_{y}^{2}(x,y,z)}}\,=\,{\frac {\mu _{0}}{4\pi }}\,\,3\mu \,{\frac {z{\sqrt {x^{2}+y^{2}}}}{(x^{2}+y^{2}+z^{2})^{5/2}}},\,$

where $\mu _{0}$ is the magnetic constant, $\pi$ is the number Pi, $\mu$ is the magnitude of ${\boldsymbol {\mu }}$, and x, y, and z are coordinates measured in metres.

A more general notation without assuming the magnetic moment ${\boldsymbol {\mu }}$ being restricted to the z direction is:

$B_{x}(x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,\,\left({\frac {3(mx+ny+pz)x}{(x^{2}+y^{2}+z^{2})^{5/2}}}-{\frac {m}{(x^{2}+y^{2}+z^{2})^{3/2}}}\right)$

$B_{y}(x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,\,\left({\frac {3(mx+ny+pz)y}{(x^{2}+y^{2}+z^{2})^{5/2}}}-{\frac {n}{(x^{2}+y^{2}+z^{2})^{3/2}}}\right)$

$B_{z}(x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,\,\left({\frac {3(mx+ny+pz)z}{(x^{2}+y^{2}+z^{2})^{5/2}}}-{\frac {p}{(x^{2}+y^{2}+z^{2})^{3/2}}}\right)$

where (m,n,p) are the components in (x,y,z) direction of the magnetic moment ${\boldsymbol {\mu }}$.

The field described above can be written in vector notation as follows:

$\mathbf {B} (x,y,z)\,=\,{\frac {\mu _{0}}{4\pi }}\,{\frac {3\mathbf {r} ({\boldsymbol {\mu }}\cdot \mathbf {r} )-{\boldsymbol {\mu }}r^{2}}{r^{5}}}$

Both the curl and the divergence of this field vanish. When more than one magnetic moment is present, the total magnetic field is simply the sum of the fields of each magnetic moment.

Effects of an external magnetic field on a magnetic moment

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

${\boldsymbol {\tau }}={\boldsymbol {\mu }}\times \mathbf {B}$

where

${\boldsymbol {\tau }}$ is the torque, measured in newton-meters,

${\boldsymbol {\mu }}$ is the magnetic moment, measured in ampere meters-squared, and

$\mathbf {B}$ is the magnetic field, measured in teslas or, equivalently in newtons per (ampere-meter).

A magnetic moment in an externally-produced magnetic field has a potential energy U:

$U=-{\boldsymbol {\mu }}\cdot \mathbf {B}$

In a case when the external magnetic field is non-uniform, there will be a force, proportional to the magnetic field gradient, acting on the magnetic moment itself. There has been some discussion on how to calculate the force acting on a magnetic dipole. There are two expressions for the force acting on a magnetic dipole. Each expression varies depending on the model used for the dipole , i.e. a current loop or two monopoles (analogous to the electric dipole). The force obtained in the case of a current loop model is

$\mathbf {F} _{l}=\nabla \left(\mathbf {m} \cdot \mathbf {B} \right)$

In the case of a pair of monopoles are used (i.e. electric dipole model)

$\mathbf {F} _{d}=\left(\mathbf {m} \cdot \ \nabla \right)\mathbf {B}$

and one can be put in terms of the other via the relation

$\mathbf {F} _{l}=\mathbf {F} _{d}+\mathbf {m} \times \left(\nabla \times \mathbf {B} \right)$

In all these expressions $\mathbf {m}$ is the dipole and $\mathbf {B}$ is the magnetic field at its position. Note that if there are no currents or time-varying electrical fields $\nabla \times \mathbf {B} =0$ and the two expressions agree.

An electron, nucleus, or atom placed in a uniform magnetic field will precess with a frequency known as the Larmor frequency. See Resonance.

Forces between two magnetic dipoles

Frames of reference for calculating the forces between two dipoles

See also: Magnetic dipole-dipole interaction

If $\mathbf {B}$ in the previous equations is replaced with the expression of the field of a magnetic dipole under the approximation for distances bigger than the characteristic length of the dipole . Namely,

$B_{x'}(\mathbf {r} )={\frac {\mu _{0}}{4\pi }}m_{1}\left({\frac {3\cos ^{2}(\theta )-1}{r^{3}}}\right)$

$B_{y'}(\mathbf {r} )={\frac {\mu _{0}}{4\pi }}m_{1}\left({\frac {3\cos(\theta )\sin(\theta )}{r^{3}}}\right)$

where the variables $r$ and $\theta$ are measured in a frame of reference with origin in $\mathbf {m} _{1}$ and oriented in such a way that $\mathbf {m} _{1}$ lies in the x-axis. This frame is called Local coordinates and is shown in the Figure on the right.

The final formulas are shown next. They are expressed in the global coordinate system,

$F_{r}(\mathbf {r} ,\alpha ,\beta )=-{\frac {3\mu _{0}}{4\pi }}{\frac {m_{2}m_{1}}{r^{4}}}\left[2\cos(\phi -\alpha )\cos(\phi -\beta )-\sin(\phi -\alpha )\sin(\phi -\beta )\right]$

$F_{\phi }(\mathbf {r} ,\alpha ,\beta )=-{\frac {3\mu _{0}}{4\pi }}{\frac {m_{2}m_{1}}{r^{4}}}\sin(2\phi -\alpha -\beta )$

In the Cartesian coordinate system, using vector notation, the above equations can be written as,

$\mathbf {F} (\mathbf {r} ,\mathbf {m} _{1},\mathbf {m} _{2})={\frac {3\mu _{0}}{4\pi r^{5}}}\left[(\mathbf {m} _{1}\cdot \mathbf {r} )\mathbf {m} _{2}+(\mathbf {m} _{2}\cdot \mathbf {r} )\mathbf {m} _{1}+(\mathbf {m} _{1}\cdot \mathbf {m} _{2})\mathbf {r} -{\frac {5(\mathbf {m} _{1}\cdot \mathbf {r} )(\mathbf {m} _{2}\cdot \mathbf {r} )}{r^{2}}}\mathbf {r} \right]$

where $\mathbf {r}$ is the distance-vector from dipole moment $\mathbf {m} _{1}$ to dipole moment $\mathbf {m} _{2}$, with $r=\|\mathbf {r} \|$, and where $\mathbf {F}$ is the force acting on $\mathbf {m} _{2}$. The force acting on $\mathbf {m} _{1}$ is in opposite direction.

The torque is straightforward to obtain from the formula

${\boldsymbol {\tau }}=\mathbf {m} _{2}\times \mathbf {B}$

which gives

${\boldsymbol {\tau }}={\frac {\mu _{0}}{4\pi }}{\frac {m_{1}m_{2}}{r^{3}}}\left[3\cos(\phi -\alpha )\sin(\phi -\beta )+\sin(\beta -\alpha )\right]$

Magnetic poles, analogy with the electric dipole moment

Magnetic 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 ${\boldsymbol {\mu }}=\mathbf {p} \mathbf {d}$, where ${\boldsymbol {\mu }}$ 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.

Although these equations are completely analogous to the case of electric dipole moment, it should be noted that magnetic dipoles are associated with angular momentum, as demonstrated by the Einstein-de Haas effect and the Barnett effect, for example. Therefore, they do not behave like ideal magnetic dipoles. In particular, although a magnetic dipole is subject to a torque in a magnetic field that tends to align its magnetic moment with the applied magnetic field, as a consequence of the associated angular momentum, the magnetic dipole precesses, that is, its direction rotates about the axis of the applied field.

See also

• Magnetism
• Magnetic dipole models
• Dipole
• Electric dipole moment
• Magnetization
• Magnetic field
• Magnetic susceptibility
• Magnetic dipole-dipole interaction
• Landau-Lifshitz-Gilbert equation