Magnetic moment
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In electromagnetism, the magnetic moment is the magnetic strength and
orientation of a magnet or other object that produces a magnetic field.
Examples of objects that have magnetic moments include loops of electric
current (such as electromagnets), permanent magnets, elementary
particles (such as electrons), various molecules, and many astronomical
objects (such as many planets, some moons, stars, etc).
More precisely, the term magnetic moment normally refers to a
system's magnetic dipole moment, the component of the magnetic moment
that can be represented by an equivalent magnetic dipole: a magnetic north
and south pole separated by a very small distance. The magnetic dipole
component is sufficient for small enough magnets or for large enough
distances. Higher-order terms (such as the magnetic quadrupole moment) may
be needed in addition to the dipole moment for extended objects.
The magnetic dipole moment of an object is readily defined in terms of the
torque that the object experiences in a given magnetic field. The same applied
magnetic field creates larger torques on objects with larger magnetic
moments. 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. The magnetic moment may be considered,
therefore, to be a vector. The direction of the magnetic moment points from
the south to north pole of the magnet (inside the 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.
Contents
Definition, units, and measurement[edit]
Definition[edit]
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:
[1]
,where τ is the torque acting on the dipole, B is the external magnetic field,
and m 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, U , with respect to external
int
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]
Units[edit]
The unit for magnetic moment in International System of Units (SI) base
units is A⋅m , where A is ampere (SI base unit of current) and m is meter (SI
2
base unit of distance). This unit has equivalents in other SI derived units
including:[3][4]
where N is newton (SI derived unit of force), T is tesla (SI derived unit of
magnetic flux density), and J is joule (SI derived unit of energy). Although
[5]
torque (N·m) and energy (J) are dimensionally equivalent, torques are never
expressed in units of energy. [6]
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:
(ESU)
(Gaussian and EMU),
, where statA is statamperes, cm is centimeters, erg is ergs, 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 formulae 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 I and area A has magnetic moment IA (see below), but in Gaussian
units the magnetic moment is IA/c.
Other units for measuring the magnetic dipole moment include the Bohr
magneton and the nuclear magneton.
Measurement[edit]
Main article: magnetometer
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[edit]
Main article: Magnetization
See also: Remanence
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 field M as:
where m and V are the magnetic dipole moment and volume of a sufficiently
ΔV ΔV
small portion of the magnet ΔV. This equation is often represented using
derivative notation such that
where dm is the elementary magnetic moment and dV is the volume element.
The net magnetic moment of the magnet m therefore is
Jump to navigationJump to search
In electromagnetism, the magnetic moment is the magnetic strength and
orientation of a magnet or other object that produces a magnetic field.
Examples of objects that have magnetic moments include loops of electric
current (such as electromagnets), permanent magnets, elementary
particles (such as electrons), various molecules, and many astronomical
objects (such as many planets, some moons, stars, etc).
More precisely, the term magnetic moment normally refers to a
system's magnetic dipole moment, the component of the magnetic moment
that can be represented by an equivalent magnetic dipole: a magnetic north
and south pole separated by a very small distance. The magnetic dipole
component is sufficient for small enough magnets or for large enough
distances. Higher-order terms (such as the magnetic quadrupole moment) may
be needed in addition to the dipole moment for extended objects.
The magnetic dipole moment of an object is readily defined in terms of the
torque that the object experiences in a given magnetic field. The same applied
magnetic field creates larger torques on objects with larger magnetic
moments. 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. The magnetic moment may be considered,
therefore, to be a vector. The direction of the magnetic moment points from
the south to north pole of the magnet (inside the 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.
Contents
Definition, units, and measurement[edit]
Definition[edit]
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:
[1]
,where τ is the torque acting on the dipole, B is the external magnetic field,
and m 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, U , with respect to external
int
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]
Units[edit]
The unit for magnetic moment in International System of Units (SI) base
units is A⋅m , where A is ampere (SI base unit of current) and m is meter (SI
2
base unit of distance). This unit has equivalents in other SI derived units
including:[3][4]
where N is newton (SI derived unit of force), T is tesla (SI derived unit of
magnetic flux density), and J is joule (SI derived unit of energy). Although
[5]
torque (N·m) and energy (J) are dimensionally equivalent, torques are never
expressed in units of energy. [6]
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:
(ESU)
(Gaussian and EMU),
, where statA is statamperes, cm is centimeters, erg is ergs, 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 formulae 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 I and area A has magnetic moment IA (see below), but in Gaussian
units the magnetic moment is IA/c.
Other units for measuring the magnetic dipole moment include the Bohr
magneton and the nuclear magneton.
Measurement[edit]
Main article: magnetometer
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[edit]
Main article: Magnetization
See also: Remanence
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 field M as:
where m and V are the magnetic dipole moment and volume of a sufficiently
ΔV ΔV
small portion of the magnet ΔV. This equation is often represented using
derivative notation such that
where dm is the elementary magnetic moment and dV is the volume element.
The net magnetic moment of the magnet m therefore is
