# How is a magnetic moment created

## Magnetic dipole moment

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

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

\$ \ vec D _ {\ vec m} = \ vec m \ times \ vec B \ ,, \$[Note 1]

by rotating it in the direction of the field (\$ \ times \$: cross product). Its potential energy is therefore dependent on the setting angle \$ \ theta \$ between field direction and magnetic moment:

\$ 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 m2. Often the product of \$ \ vec {m} \$ and the magnetic field constant \$ \ mu_0 \$ is used (see note 1); this has the SI unit T · m3.

### Come about

The current density distribution \$ \ vec {\ jmath} \, (\ vec r) \$ has a magnetic moment
\$ \ vec {m} = \ frac {1} {2} \ int \ limits _ {\ mathbb {R} ^ 3} \ mathrm {d} ^ 3r \ left [\ vec {r} \ times \ vec {\ jmath } \, (\ vec {r}) \ right]. \$
This results in a flat current loop
\$ m = I \ cdot A, \$
where \$ A \$ is the area surrounded by the current \$ I \$.
In electrical engineering, this is the basis for z. B. Generators, motors and electromagnets.
• Particles with an intrinsic angular momentum (spin) \$ \ vec s \$ have a magnetic moment
\$ \ vec {m} = \ gamma \ vec {s}. \$
\$ \ 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.

### Level conductor loop

The following applies to a closed conductor loop

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

Here designated

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

Thus it follows for the magnetic dipole moment:

\$ \ 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 \$ \ vec {n} _A \$ on the flat surface \$ A \$.

#### Long coil with current flowing through it

The magnetic moment of a coil with current flowing through it is the product of the number of turns \$ n \$, current strength \$ I \$ and area \$ A \$:

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

Here \$ \ vec {A} = \ vec {n} _A A \$ is the vector belonging to the area \$ A \$.

See also: magnetic interlinking flux

### Charged particle on a circular path

#### Classic

If the circular current is caused by a particle with the mass \$ M \$ and the charge \$ Q \$ circling on a circular path (radius \$ r \$, period of rotation \$ T \$), this formula results

\$ \ 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

\$ \ vec {L} = \ omega M r ^ 2 \; \ vec {n} _A \$

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

#### 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 integer multiples of the unit \$ \ hbar \$ (Planck's quantum of action) [Note 2], also has the magnetic one Orbit moment a smallest unit, the magneton:

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

If the elementary charge \$ e \$ is substituted for \$ Q \$, the result for the electron is the Bohr magneton \$ \ mu_B = \ tfrac {e \ hbar} {2m_e} \$, for the proton the nuclear magneton \$ \ mu_K = \ tfrac { e \ hbar} {2m_p} \$. Since the proton mass \$ m_p \$ is almost 2000 times greater than the electron mass \$ m_e \$, the nuclear magneton is smaller than Bohr's magneton by the same factor. Therefore, the magnetic effects of the atomic nuclei are very weak and difficult to observe (hyperfine structure).

### The magnetic moment of particles and nuclei

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

\$ \ vec \ mu_s = g_e \, \ mu_B \, \ frac {\ vec {s}} {\ hbar} \$ with Bohr's magneton \$ \ mu _B \$,

for proton (p) and neutron (n)

\$ \ vec \ mu _s = g_ {p, n} \, \ mu_K \, \ frac {\ vec {s}} {\ hbar} \$ with the nuclear magneton \$ \ mu _K \$,

and analogously 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.[Note 3]

Particle Spin-g factor
Electron \$ e ^ - \$\$ -2{,}002\,319\,304\,361\,82(52) \$
Muon \$ \ mu ^ - \$\$ -2{,}002\,331\,8418(13) \$
Proton \$ p \$\$ +5{,}585\,694\,702(17) \$
Neutron \$ n \$\$ -3{,}826\,085\,45(90) \$

The numbers in brackets indicate the estimated standard deviation.

According to the Dirac theory, the Landé factor of the fundamental fermions is exactly \$ g_s \ mathord = \ pm 2 \$, quantum electrodynamically a value of about \$ g_s \ mathord = \ pm 2 {,} 0023 \$ 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.

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

\$ \ vec {\ mu} = \ vec {\ mu} _s + \ vec {\ mu} _ \ ell \$.

### Magnetic field of a magnetic dipole

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

\$ \ vec {B} (\ vec {r}) \, = \, \ frac {\ mu_0} {4 \ pi} \, \ frac {3 \ vec {r} (\ vec {m} \ cdot \ vec {r}) - \ vec {m} r ^ 2} {r ^ 5} \$.

Here \$ \ mu_0 \$ 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

\$ \ vec {A} (\ vec {r}) \, = \, \ frac {\ mu_0} {4 \ pi} \, \ frac {\ vec {m} \ times \ vec {r}} {r ^ 3} \$,

where \$ \ vec {B} = \ nabla \ times \ vec {A} \$.

### Force effect between two dipoles

The force exerted by dipole 1 on dipole 2 is

\$ \ vec {F} = \ nabla \ left (\ vec {m} _2 \ cdot \ vec {B} _1 \ right) \$

It surrenders

\$ \ 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 \$ \ vec {r} _n \$ is the unit vector that points from dipole 1 to dipole 2 and \$ r \$ is the distance between the two magnets. The force on dipole 1 is reciprocal.

### Torque effect between two dipoles

The torque exerted by dipole 1 on dipole 2 is

\$ \ vec {M} = \ vec {m} _2 \ times \ vec {B} _1 \$

where \$ \ vec {B} _1 \$ is the field generated by dipole 1 at the location of dipole 2 (see above). The torque on dipole 1 is reciprocal.

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, Göschen Collection, Volume II (Electrodynamics), is called magnetic moment defines the \$ \ mu_0 \$ -fold of the value given here. Then it says z. B. \$ \ vec D _ {\ vec m} = \ vec m \ times \ vec H \$ and \$ \ vec m \$ is not defined as magnetization through volume, but as magnetic polarization \$ \ vec J \, \, (= \ mu_0 \ vec M) \$ by volume. In matter there is generally \$ \ vec B = \ mu_0 \ cdot \ vec H + \ vec J \$ and \$ \ vec m \ times \ vec J \ equiv 0 \$ (because of \$ \ vec M \ times \ mu_0 \, \ vec M \ equiv 0 \,. \$) Old and new definitions are therefore fully equivalent. The official agreement on the new CODATA definition did not take place until 2010.
2. ↑ More precisely: this applies to the component of the angular momentum vector along an axis.
3. ↑ Exactly the sign only has to be taken into account when it comes to the direction of rotation of the Larmor precession or the sign of the paramagnetic spin polarization. Therefore, the signs are not handled in a completely uniform way in the literature.
1. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 28, 2015. Value for the g-factor of the electron
2. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 28, 2015. Value for the muon g-factor
3. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 28, 2015. Value for the g-factor of the proton
4. CODATA Recommended Values. National Institute of Standards and Technology, accessed July 28, 2015. Value for the g-factor of the neutron