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 (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.
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.See, for example, where the relevant is .
Units The unit for magnetic moment in International System of Units (SI) base units is A⋅m2, where A is ampere (SI base unit of current) and m is meter (SI base unit of distance). This unit has equivalents in other SI derived units including:
where N is newton (SI unit of force), T is tesla (SI unit of magnetic flux density), and J is joule (SI unit of energy).
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 (non-equivalent) 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 (EMU/ESU) 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 (see below), 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 (where both the magnitude and the direction of is the same for the entire magnet (such as a straight bar magnet) 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 (or remanence), denoted . The formula needed in this case to calculate in (units of A⋅m2) is:
where: is the residual flux density, expressed in teslas. is the volume of the magnet (in m3). is the permeability of vacuum ().
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.
Force on a moment
A magnetic moment in an externally produced magnetic field has a potential energy :
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 are two expressions for the force acting on a magnetic dipole, depending on whether the model used for the dipole is a current loop or two monopoles (analogous to the electric dipole). 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 is And one can be put in terms of the other via the relation
In all these expressions is the dipole and is the magnetic field at its position. If there are no currents or time-varying electrical fields or magnetic charge, , and the two expressions agree.
Relation to free energy
One can relate the magnetic moment of a system to the free energy of that system. In a uniform magnetic field , the free energy can be related to the magnetic moment of the system as
where is the entropy of the system and is the temperature. Therefore, the magnetic moment can also be defined in terms of the free energy of a system as
Magnetism
In addition, an applied magnetic field can change the magnetic moment of the object itself; for example by magnetizing it. This phenomenon is known as magnetism. An applied magnetic field can flip the magnetic dipoles that make up the material causing both paramagnetism and ferromagnetism. Additionally, the magnetic field can affect the currents that create the magnetic fields (such as the atomic orbits) which causes diamagnetism.
Effects on environment
Magnetic field of a magnetic moment
Any 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 higher-order multipole 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 the magnetic 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 of μ0 = , where μ0 is known as the vacuum permeability. For example:
Forces between two magnetic dipoles
As discussed earlier, the force exerted by a dipole loop with moment on another with moment is
where is the magnetic field due to moment . The result of calculating the gradient is
where is the unit vector pointing from magnet 1 to magnet 2 and is the distance. An equivalent expression is
The force acting on is in the opposite direction.
Torque of one magnetic dipole on another The torque of magnet 1 on magnet 2 is
Theory underlying magnetic dipoles
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 the monopole (represented by an isolated magnetic north or south pole) the dipole (represented by two equal and opposite magnetic poles), and the quadrupole (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.
For many magnets the first non-zero term is the magnetic dipole moment. (To date, no isolated magnetic 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).
Magnetic potentials
Traditionally, the equations for the magnetic dipole moment (and higher order terms) are derived from theoretical quantities called magnetic potentials which are simpler to deal with mathematically than the magnetic fields.
In the magnetic pole model, the relevant magnetic field is the demagnetizing field . Since the demagnetizing portion of does not include, by definition, the part of due to free currents, there exists a magnetic 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 a magnetic 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: