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Magnetic moment

In physics, the magnetic moment of an object is a vector property, denoted here as m, that determines the torque, denoted here by τ, it experiences in a magnetic flux density B, namely τ = m × B (where × denotes the vector cross product). As such, it also determines the change in potential energy of the object, denoted here by U, when it is introduced to this flux, namely U = −m·B.

Origin

A magnetic moment may have a macroscopic origin in a bar magnet or a current loop, for example, or microscopic origin in the spin of an elementary particle like an electron, or in the angular momentum of an atom.

Macroscopic examples

The electric motor is based upon the torque experienced by a current loop in a magnetic field. The basic idea is that the current in the loop is made up of moving electrons, which are subect to the Lorentz force F in a magnetic field:

$\mathbf {F} =-e\left(\mathbf {v\times B} \right)\ ,$

where e is the electron charge and v is the electron velocity. This force upon the electrons is communicated to the wire loop because the electrons cannot escape the wire, and so exert a force upon it. The electrons at the top of the loop move oppositely to those at the bottom, so the force at the top is opposite in direction to that at the bottom. If the magnetic field is in the plane of the loop, the forces are normal to this plane, causing a torque upon the loop tending to turn the loop about an axis along the direction of the field.

The torque exerted upon a current loop of radius a carrying a current I, placed in a uniform magnetic flux density B at an angle to the unit normal û_n to the loop is:

${\boldsymbol {\tau }}={\mathit {I}}\mathbf {S\times B} \ ,$

where the vector S is:

$\mathbf {S} =\pi a^{2}\ {\hat {\mathbf {u} }}_{n}\ .$

Consequently the magnetic moment of this loop is:

$\mathbf {m} ={\mathit {I}}\ \mathbf {S} \ .$

Microscopic examples

At a fundamental level, magnetic moment is related to the angular momentum of fundamental particles. In this discussion, focus is upon the electron and the atom.

The discussion splits naturally into two parts: kinematic and dynamic.

Kinematics

The kinematical discussion, which does not enter upon the physical origins of magnetism and its effects upon mechanics, deals with the classification of atomic states based upon symmetry. Although these ideas apply to nucleii and other particles, here attention is focused on electrons in atoms. The symmetry analysis leads to the identification of spin S and orbital angular momentum L and its combination J = L + S.

The electron has a spin. The resultant total spin S of an ensemble of electrons in an atom is the vector sum of the constituent spins s_j:

$\mathbf {S} =\sum _{j=1}^{N}\ \mathbf {s_{j}} \ .$

Likewise, the orbital momenta of an ensemble of electrons in an atom add as vectors.

Where both spin and orbital motion are present, they combine by vector addition:

$\mathbf {J=L+S} \ .$

Dynamics

The dynamic aspect introduces the proportionality between magnetic moment and angular momentum, the gyromagnetic ratio, and attempts to explain its origin based upon quantum electrodynamics. The magnetic moment m_S of a system of electrons with spin S is:

$\mathbf {m_{S}} =2m_{B}\mathbf {S} \ ,$

and the magnetic moment m_L of an electronic orbital momentum L is:

$\mathbf {m_{L}} =m_{B}\mathbf {L} \ .$

Here the factor m_B refers to the Bohr magneton, defined by:

$m_{B}={\frac {e\hbar }{2m_{e}}}\ ,$

with e = the electron charge, ℏ = Planck's constant divided by 2π, and m_e = the electron mass. These relations are generalized using the g-factor:

$\mathbf {m_{J}} =gm_{B}\ \mathbf {J} \ ,$

with g=2 for spin (J = S) and g=1 for orbital motion (J = L).

The magnetic moment of an atom of angular momentum J is

$\mathbf {m_{J}} =gm_{B}\mathbf {J} \ ,$

with g now the Landé g-factor or spectroscopic splitting factor:

$g={\frac {3}{2}}+{\frac {S(S+1)-L(L+1)}{2J(J+1)}}\ .$

If an atom with this associated magnetic moment now is subjected to a magnetic flux, it will experience a torque due to the applied field.