Magnetic moment

, The magnetic moment ( and magnetic dipole moment ) is a measure of the strength of a magnetic dipole and electric dipole moment defined analogously to in physics.

A magnetic moment acting in an external magnetic field of flux density, a torque

By which it is rotated into the field direction (: cross product ). Its potential energy is therefore dependent on the setting angle between the field direction and the magnetic moment:

Important examples are the compass needle and the electric motor.

The unit of the magnetic moment is in the SI system A · m2, multiplied by the magnetic field constant it is T · m3.

A magnetic moment can have two causes:

• An electric current having the current density distribution has a magnetic moment

• Particles with an intrinsic angular momentum (spin ) have a magnetic moment

• 4.1 Mechanical force between two dipoles
• 4.2 Torque effect between two dipoles

Examples

Plane conductor loop

For a closed conductor loop

It referred

• The current density at the location of
• A volume integral
• The current through the conductor loop
• A path integral along the conductor loop.

This yields for the magnetic dipole moment:

With the normal vector to the plane surface.

Current-carrying long coil

The magnetic moment of a current-carrying coil is the product of the number of turns, current and area:

See also: magnetic flux linkage

Charged particles along a circular path

Classic

Is the circulating current caused by a charge on a circular path (radius, orbital period ), this formula gives

The magnetic moment is so tight with the angular momentum

Linked. The constant factor, in the mass of the particle, is the gyromagnetic ratio for moving cargo along the circular path. ( When converting the angular velocity is used.)

Quantum mechanically

The classic formula plays in atomic and nuclear physics an important role, because it is also true in quantum mechanics, and a well-defined angular momentum associated with each energy level of a single atom or nucleus. Since the angular momentum of the spatial motion ( orbital angular momentum, in contrast to the spin) only integer multiples of the unit ( Planck's constant ) can take [Note 2], and this orbital magnetic moment has a smallest unit, the magneton:

For this is called the electron Bohr magneton, as for the proton nuclear magnetons. Since the proton mass is about 2000 times larger than the electron mass, the nuclear magneton is the same factor smaller than the Bohr magneton. Therefore, the magnetic effects of the atomic nuclei are very weak and difficult to observe ( hyperfine structure).

The magnetic moment of the particles and cores

Particles and atomic nuclei with a spin possess a magnetic spin moment, which is to their spin parallel (or antiparallel), but in relation to the spin has a different size, as if it proceeded from an equally large orbital angular momentum. This is expressed by the abnormal Landé factor of the spin. One writes for electron ( ) and positron ( )

For proton (p) and neutron ( n )

Analog and other particles. For the muon the muon Bohr magneton is in place of the mass of the electron used for the quarks their respective constituent mass and drittelzahlige electrical charge. Is the magnetic moment in antiparallel to the spin, the g-factor is negative. However, this sign convention [Note 3] is not applied consistently, so that often the g-factor is given, for example, of the electron as positive.

According to the Dirac theory of the Landé factor of the fundamental fermions is exact, quantum electrodynamic a value of approximately predicted. Precise measurements of electron or positron and the muon are agreeing excellent agreement, including the predicted small difference between the electron and muon, thus confirming the Dirac theory and quantum electrodynamics. The strongly divergent g - factors for the nucleons are, however, be explained by differences in the percent range, due to their structure from three constituent quarks.

, The particles (eg, electrons, which are bonded to an atomic nucleus ) in addition an orbital angular momentum, so is that the magnetic moment of the above- considered magnetic moment of the spins ( ) and the orbital angular momentum () is composed:

The magnetic field of a magnetic dipole

A magnetic dipole at the origin at the location of leading to a magnetic flux density

It is the magnetic field constant. Except at the origin, where the field diverges vanishes throughout both the rotation and the divergence of this field. The corresponding vector potential is given by

Being.

Force and moment interaction between magnetic dipoles

Force between two dipoles

The force exerted on the dipole of the dipole 1 is 2,

The result is

Wherein the unit vector which points from one dipole to the dipole 2 and the distance between the two magnets. The force on dipole 1 is reciprocal.

Torque action between two dipoles

The torque which is exerted on one of dipole dipole 2 is

Where the field generated by a dipole 1 at the location of dipole 2 (see above). The torque on dipole 1 is reciprocal.

In the presence of a plurality of dipoles, the forces or moments can be superimposed. Since soft magnetic materials form a field-dependent dipole, these equations are not applicable.