Magnetic dipole moment

Physical size
Surname	Magnetic dipole moment
Formula symbol	${\displaystyle {\vec {m}},{\vec {\mu }}}$
Size and unit system unit dimension SI A · m 2 I · L 2 Gauss ( cgs ) erg / Gs = ABA · cm 2 L 02/05 · M 02/01 · T -1 esE ( cgs ) Stata · cm 2 L 02/03 · M 01/02 · T -2 emE ( cgs ) erg / Gs = ABA · cm 2 L 02/05 · M 02/01 · T -1

The magnetic dipole moment (or magnetic moment ) is a measure of the strength of a magnetic dipole in physics and is defined analogously to the electric dipole moment . ${\displaystyle {\vec {m}}}$

A torque acts on a magnetic dipole in an external magnetic field of the flux density ${\displaystyle {\vec {B}}}$

${\displaystyle {\vec {D}}_{\vec {m}}={\vec {m}}\times {\vec {B}}\,,}$ 

by which it is rotated in the field direction ( : cross product ). Its potential energy therefore depends on the setting angle between the field direction and the magnetic moment: ${\displaystyle \times }$ ${\displaystyle \theta }$

${\displaystyle E_{\text{pot}}=-{\vec {m}}\cdot {\vec {B}}\equiv -m\,B\cos \theta .}$

Important examples are the compass needle and the electric motor .

The unit of measurement of the magnetic moment in the International System of Units (SI) is A · m ² . Often the product of and the magnetic field constant is used (see note 1); this has the SI unit T · m ³ . ${\displaystyle {\vec {m}}}$ ${\displaystyle \mu _{0}}$

occurrence

• A magnetic moment can result from an electric current :

The current density distribution has a magnetic moment ${\displaystyle {\vec {\jmath }}\,({\vec {r}})}$

${\displaystyle {\vec {m}}={\frac {1}{2}}\int \limits _{\mathbb {R} ^{3}}\mathrm {d} ^{3}r\left[{\vec {r}}\times {\vec {\jmath }}\,({\vec {r}})\right].}$

This results in a flat current loop

${\displaystyle m=I\cdot A}$

where is the area around which the current flows . ${\displaystyle A}$ ${\displaystyle I}$

In electrical engineering, this is the basis for z. B. Generators , motors and electromagnets .

• Particles with their own angular momentum ( spin ) have a magnetic moment ${\displaystyle {\vec {s}}}$

${\displaystyle {\vec {m}}=\gamma {\vec {s}}.}$

${\displaystyle \gamma }$is called the gyromagnetic ratio . Examples are electrons , which cause the ferromagnetism of the elements of the iron group and the rare earths by the parallel positioning of their magnetic moments . Ferromagnetic materials are used as permanent magnets or as iron cores in electromagnets and transformers .

Examples

Level conductor loop

The following applies to a closed conductor loop

${\displaystyle \int {\vec {\jmath }}\,({\vec {r}})\;\mathrm {d} ^{3}r=\int I\;\mathrm {d} {\vec {r}}.}$

Here designated

Magnetic dipole moment of an area around which current flows

• ${\displaystyle {\vec {\jmath }}\,({\vec {r}})}$ the current density at the location ${\displaystyle {\vec {r}}}$
• ${\displaystyle \int \mathrm {d} ^{3}r}$a volume integral
• ${\displaystyle I}$the current through the conductor loop
• ${\displaystyle \int \mathrm {d} {\vec {r}}}$a path integral along the conductor loop.

Thus it follows for the magnetic dipole moment:

${\displaystyle {\vec {m}}={\frac {I}{2}}\int _{C}({\vec {r}}\times \mathrm {d} {\vec {r}})=I\cdot {\vec {A}}=I\cdot A\cdot {\vec {n}}_{A}}$

with the normal vector on the flat surface . The vector is oriented in such a way that it points upwards when the current is flowing counterclockwise . ${\displaystyle {\vec {n}}_{A}}$ ${\displaystyle A}$ ${\displaystyle {\vec {n}}_{A}}$

Long coil with current flowing through it

The magnetic moment of a current-carrying coil is the product of the number of turns , current strength and area : ${\displaystyle n}$ ${\displaystyle I}$ ${\displaystyle A}$

${\displaystyle {\vec {m}}=n\cdot I\cdot {\vec {A}}.}$

This contains the vector belonging to the area . ${\displaystyle {\vec {A}}={\vec {n}}_{A}A}$ ${\displaystyle A}$

See also : magnetic interlinking flux

Charged particle on a circular path

Classic

If the circulating current is caused by the fact that a particle with its mass and charge orbits on a circular path (radius , period of revolution ), this formula results ${\displaystyle M}$ ${\displaystyle Q}$ ${\displaystyle r}$ ${\displaystyle T}$

${\displaystyle {\vec {m}}=IA\;{\vec {n}}_{A}={\frac {Q}{T}}\cdot \pi r^{2}\;{\vec {n}}_{A}\equiv {\frac {Q}{2M}}{\vec {L}}\quad .}$

The magnetic moment is therefore fixed with the angular momentum

${\displaystyle {\vec {L}}=\omega Mr^{2}\;{\vec {n}}_{A}}$

connected. The constant factor is the gyromagnetic ratio for moving charges on the circular path. (The angular velocity is used for the conversion .) ${\displaystyle \gamma ={\tfrac {Q}{2M}}}$ ${\displaystyle \omega ={\tfrac {2\pi }{T}}}$

Quantum mechanics

The classical formula plays a major role in atomic and nuclear physics, because it also applies in quantum mechanics, and a well-defined angular momentum belongs to every energy level of an individual atom or nucleus. Since the angular momentum of the spatial movement ( orbital angular momentum , in contrast to the spin ) can only be whole-number multiples of the constants ( Planck's quantum of action ), the magnetic orbital moment also has a smallest "unit", the magneton : ${\displaystyle \hbar }$

${\displaystyle \mu ={\frac {Q\hbar }{2M}}\quad .}$

Is for the elementary charge used, the result for the electron, the Bohr magneton , for the proton , the nuclear magneton . Since the proton mass is almost 2000 times larger than the electron mass , the nuclear magneton is smaller than Bohr's magneton by the same factor. Therefore, the magnetic effects of the atomic nuclei based on the nuclear magneton and the resulting ( hyperfine structure ) are very weak and difficult to observe in comparison with the fine structure splitting based on the Bohr magneton . ${\displaystyle Q}$ ${\displaystyle e}$ ${\displaystyle \mu _{\mathrm {B} }={\tfrac {e\hbar }{2m_{\mathrm {e} }}}}$ ${\displaystyle \mu _{\mathrm {K} }={\tfrac {e\hbar }{2m_{\mathrm {p} }}}}$ ${\displaystyle m_{\mathrm {p} }}$ ${\displaystyle m_{\mathrm {e} }}$ ${\displaystyle \mu _{\mathrm {K} }}$ ${\displaystyle \mu _{\mathrm {B} }}$

The magnetic moment of particles and nuclei

Particles and atomic nuclei with one spin have a magnetic spin moment that is parallel (or anti-parallel) to their spin, but has a different size in relation to the spin than if it came from an orbital angular momentum of the same size. This is expressed by the anomalous Landé factor of the spin . One writes for electron ( ) and positron ( ) ${\displaystyle {\vec {s}}}$ ${\displaystyle {\vec {\mu }}_{s}}$ ${\displaystyle g_{s}{\mathord {\neq }}1}$ ${\displaystyle \mathrm {e} ^{-}}$ ${\displaystyle \mathrm {e} ^{+}}$

${\displaystyle {\vec {\mu }}_{s}=g_{e}\,\mu _{\mathrm {B} }\,{\frac {\vec {s}}{\hbar }}}$with Bohr's magneton , ${\displaystyle \mu _{\mathrm {B} }}$

for proton (p) and neutron (n)

${\displaystyle {\vec {\mu }}_{s}=g_{p,n}\,\mu _{\mathrm {K} }\,{\frac {\vec {s}}{\hbar }}}$with the nuclear magneton , ${\displaystyle \mu _{\mathrm {K} }}$

and correspondingly for other particles. For the muon, in Bohr's magneton, instead of the mass of the electron, that of the muon is used, for the quarks their respective constituent mass and third-digit electrical charge. If the magnetic moment is antiparallel to the spin, the g-factor is negative. However, this sign convention is not applied consistently, so that the g-factor z. B. the electron is indicated as positive.

Particle	Spin-g factor
electron ${\displaystyle e^{-}}$	${\displaystyle -2{,}002\,319\,304\,362\,56(35)}$
Muon ${\displaystyle \mu ^{-}}$	${\displaystyle -2{,}002\,331\,8418(13)}$
proton ${\displaystyle p}$	${\displaystyle +5{,}585\,694\,6893(16)}$
neutron ${\displaystyle n}$	${\displaystyle -3{,}826\,085\,45(90)}$

The numbers in brackets indicate the estimated standard deviation .

According to Dirac's theory , the Landé factor of the fundamental fermions is exact , quantum electrodynamically a value of about is predicted. Precise measurements on the electron or positron as well as on the muon are in excellent agreement, including the predicted small difference between electron and muon, and thus confirm the Dirac theory and quantum electrodynamics. The strongly deviating g-factors for the nucleons can be explained by their structure of three constituent quarks, albeit with deviations in the percentage range . ${\displaystyle g_{s}{\mathord {=}}\pm 2}$ ${\displaystyle g_{s}{\mathord {=}}\pm 2{,}0023}$

If the particles (e.g. electrons bound to an atomic nucleus) also have an orbital angular momentum, the magnetic moment is the sum of the magnetic moment of the spin considered above and that of the orbital angular momentum: ${\displaystyle {\vec {\mu }}_{s}}$ ${\displaystyle {\vec {\mu }}_{\ell }}$

${\displaystyle {\vec {\mu }}={\vec {\mu }}_{s}+{\vec {\mu }}_{\ell }}$.

Magnetic field of a magnetic dipole

A magnetic dipole at the origin of the coordinates leads to a magnetic flux density at the location ${\displaystyle {\vec {m}}}$ ${\displaystyle {\vec {r}}}$

${\displaystyle {\vec {B}}({\vec {r}})\,=\,{\frac {\mu _{0}}{4\pi }}\,{\frac {3{\vec {r}}({\vec {m}}\cdot {\vec {r}})-{\vec {m}}r^{2}}{r^{5}}}}$.

This is the magnetic field constant . Except at the origin, where the field diverges, everywhere both the rotation and the divergence of this field vanish . The associated vector potential is given by ${\displaystyle \mu _{0}}$

${\displaystyle {\vec {A}}({\vec {r}})\,=\,{\frac {\mu _{0}}{4\pi }}\,{\frac {{\vec {m}}\times {\vec {r}}}{r^{3}}}}$,

where is. With the magnetic field strength , the magnetic scalar potential is ${\displaystyle {\vec {B}}=\nabla \times {\vec {A}}}$ ${\displaystyle {\vec {H}}=-\nabla \psi }$

${\displaystyle \psi ({\vec {r}})\,=\,{\frac {1}{4\pi }}\,{\frac {{\vec {m}}\cdot {\vec {r}}}{r^{3}}}}$.

Force and moment effects between magnetic dipoles

See also : dipole force

Force effect between two dipoles

The force exerted by dipole 1 on dipole 2 is

${\displaystyle {\vec {F}}=\nabla \left({\vec {m}}_{2}\cdot {\vec {B}}_{1}\right)}$

It turns out

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

where is the unit vector pointing from dipole 1 to dipole 2 and is the distance between the two magnets. The force on dipole 1 is reciprocal. ${\displaystyle {\vec {r}}_{n}}$ ${\displaystyle r}$

Torque effect between two dipoles

The torque exerted by dipole 1 on dipole 2 is

${\displaystyle {\vec {M}}={\vec {m}}_{2}\times {\vec {B}}_{1}}$

where the field generated by dipole 1 is at the location of dipole 2 (see above). The torque on dipole 1 is reciprocal. ${\displaystyle {\vec {B}}_{1}}$

In the presence of several dipoles, the forces or moments can be superimposed. Since soft magnetic materials form a field-dependent dipole, these equations cannot be used.

literature

• John David Jackson : Classical Electrodynamics . Appendix about units and dimensions. 4th edition. De Gruyter, Berlin 2006, ISBN 3-11-018970-4 .

Remarks

1. ↑ In older books, e.g. B. W. Döring , Introduction to Theoretical Physics , collection Göschen, Band II (electrodynamics) is, as a magnetic moment , the defined times the value given here. Then it says z. B. and is defined not as magnetization through volume, but as magnetic polarization through volume. In matter is general and (because of ) the old and new definitions are therefore fully equivalent. The official agreement on the new CODATA definition did not come until 2010. ${\displaystyle \mu _{0}}$ ${\displaystyle {\vec {D}}_{\vec {m}}={\vec {m}}\times {\vec {H}}}$ ${\displaystyle {\vec {m}}}$ ${\displaystyle {\vec {J}}\,\,(=\mu _{0}{\vec {M}})}$ ${\displaystyle {\vec {B}}=\mu _{0}\cdot {\vec {H}}+{\vec {J}}}$ ${\displaystyle {\vec {m}}\times {\vec {J}}\equiv 0}$ ${\displaystyle {\vec {M}}\times \mu _{0}\,{\vec {M}}\equiv 0\,.}$
2. ↑ More precisely: this applies to the component of the angular momentum vector along an axis.
3. ↑ The sign is only of practical importance when it comes to the direction of rotation of the Larmor precession or the sign of the paramagnetic spin polarization . Accordingly, the signs are not handled in a completely uniform manner in the literature.

1. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed July 21, 2019 .  Value for the g-factor of the electron
2. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed July 21, 2019 .  Value for the g-factor of the muon
3. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed July 21, 2019 .  Value for the g-factor of the proton
4. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed July 21, 2019 .  Value for the g-factor of the neutron