Magnetic moment

In electromagnetism, the magnetic moment or magnetic dipole moment is a vector quantity which characterizes the strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude of torque 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 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.
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.
Example magnetic moments for subatomic particles include electron magnetic moment, nuclear magnetic moment, and nucleon magnetic moment.

Definition, units, and measurement

Definition

The magnetic moment 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 is the torque acting on the dipole, is the external magnetic field, and 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 for thermodynamics calculations of the magnetic moment. In this definition, the magnetic dipole moment of a system is the negative gradient of its intrinsic energy,, 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.

Units

The unit for magnetic moment in International System of Units base units is A⋅m², where A is ampere and m is meter. This unit has equivalents in other SI derived units including:
where N is newton, T is tesla, and J is joule.
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 alternative units of magnetic dipole moment:
where statA is statampere, cm is centimeter, erg is erg, and G is gauss. The ratio of these two non-equivalent CGS units is equal to the speed of light in free space, expressed in cm⋅s⁻¹.
All formulas in this article are correct in SI units; they may need to be changed for use in other unit systems. For example, in SI units, a loop of current with current and area has magnetic moment , but in Gaussian units the magnetic moment is.
Other units for measuring the magnetic dipole moment include the Bohr magneton and the nuclear magneton.

Measurement

The magnetic moments of objects are typically measured with devices called magnetometers, though not all magnetometers measure magnetic moment: Some are configured to measure magnetic 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.

Relation to magnetization

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 magnetization as:
where and are the magnetic dipole moment and volume of a sufficiently small portion of the magnet This equation is often represented using derivative notation such that
where is the elementary magnetic moment and is the volume element. The net magnetic moment of the magnet therefore is
where the triple integral denotes integration over the volume of the magnet. For uniform magnetization the last equation simplifies to:
where is the volume of the bar magnet.
The magnetization is often not listed as a material parameter for commercially available ferromagnetic materials, though. Instead the parameter that is listed is residual flux density, denoted . The formula needed in this case to calculate in is:
where:

is the residual flux density, expressed in teslas.
is the volume of the magnet.
is the permeability of vacuum.
Models

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. In magnetic materials, the cause of the magnetic moment are the [|spin and orbital angular momentum states] of the electrons, and varies depending on whether atoms in one region are aligned with atoms in another.

Magnetic pole model

The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. This is sometimes known as the Gilbert model. In this model, a small magnet is modeled by a pair of fictitious magnetic monopoles 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: the strength of its poles, and the vector separating them. The magnetic dipole moment is related to the fictitious poles as
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 with angular momentum. Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets. Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field, in analogy to the electric field.

Ampèrian loop model

After Hans Christian Ørsted discovered that electric currents produce a magnetic field and André-Marie Ampère discovered that electric 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 current I. The dipole moment of this loop is
where 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.

Localized current distributions

The magnetic dipole moment can be calculated for a localized current distribution assuming that we know all of the currents involved. Conventionally, the derivation starts from a multipole expansion of the vector potential. This leads to the definition of the magnetic dipole moment as:
where is the vector cross product, is the position vector, and is the electric current density and the integral is a volume integral. When the current density in the integral is replaced by a loop of current I in a plane enclosing an area S then the volume integral becomes a line integral and the resulting dipole moment becomes
which is how the magnetic dipole moment for an Amperian loop is derived.
Practitioners using the current loop model generally represent the magnetic field by the solenoidal field, analogous to the electrostatic field.

Magnetic moment of a solenoid

A generalization of the above current loop is a coil, or solenoid. Its moment is the vector sum of the moments of individual turns. If the solenoid has identical turns and vector area,

Quantum mechanical model

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 the angular momentum and the magnetic moment of a particle. While this relation is straightforward to develop for macroscopic currents using the amperian loop model, neither the magnetic pole model nor the amperian loop model truly represents what is occurring at the atomic and molecular levels. At that level quantum 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 of the particle and the particle's orbital angular momentum. See [|below] for more details.

Effects of an external magnetic field

Torque on a moment

The torque on an object having a magnetic dipole moment in a uniform magnetic field 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.
An electron, nucleus, or atom placed in a uniform magnetic field will precess with a frequency known as the Larmor frequency. See Resonance.