 Where are magnetic dipoles coming from? The classical answer is simple:
A magnetic moment m is generated whenever a current flows in closed circle. 
  Of course, we will not mix
up the letter m used for magnetic moments with the m*_{e} , the mass of an electron, which we
also need in some magnetic equations. 
  For a current I flowing in a circle enclosing an area A, m is
defined to be 
 

  This does not only apply to
"regular" current flowing in a wire, but in the extreme also to a single electron
circling around an atom. 
 In the context
of Bohrs model for an atom,
the magnetic moment of such an electron is easily understood: 
  The current I carried by one electron orbiting the nucleus at the distance r with the frequency
n = w/2p is . 
 

  The area A is
p r^{2}, so we have for the magnetic moment
m_{orb} of the electron 
 
m_{orb}  = 
e ·  w 2p  · p
r^{2}  =  ½ · e · w · r ^{2} 


 Now the mechanical angular momentum L is given by 
 

  With
m*_{e} = mass of electron (the * serves to distinguish the mass
m*_{e} from the magnetic moment m^{e} of the electron), and we have a
simple relation between the mechanical angular momentum L of an electron (which, if you remember, was the decisive quantity in the Bohr atom
model) and its magnetic moment m. 
 
m_{orb} = –  e 2m*_{e}  · L 


  The minus sign takes into account that mechanical angular momentum and magnetic moment are
antiparallel; as before we note that this is a vector equation because both
m and L are (polar) vectors. 
  The quantity e/2m*_{e} is called the
gyromagnetic relation or quotient; it should be a
fixed constant relating m and any given L. 
  However, in real life it often deviates from the
value given by the formula. How can that be? 
  Well, try to remember: Bohr's model is a mixture of classical physics and quantum physics and far too simple to account for everything. It is thus small wonder that conclusions based on
this model will not be valid in all situations. 
 In
proper quantum mechanics (as in Bohr's semiclassical model) L comes
in discrete values only. In particular, the fundamental assumption of Bohr's model
was L = n · , with n = quantum number = 1, 2, 3, 4,
... 
  It follows that
m_{orb} must be quantized, too; it must come in multiples of

 
m_{orb} =  h · e _{ } 4p
· m*_{e}  = m_{Bohr} = 9.27
· 10^{–24} Am^{2} 


  This relation defines a fundamental unit for magnetic dipole
moments, it has its own name and is called a Bohr magneton. 
  It is for magnetism what an
elementary charge is for electric effects. 
 But electrons
orbiting around a nucleus are not the only source of magnetic moments. 
  Electrons always have a spin s, which, on the level of the Bohr model, can be
seen as a builtin angular momentum with the value · s. The spin quantum number
s is ½, and this allows two directions of angular momentum and magnetic moment , always
symbolically written as . 
 

 It is possible, of course, to compute
the circular current represented by a charged ball spinning around its axis if the distribution of charge in the
sphere (or on the sphere), is known, and thus to obtain the magnetic moment of the spinning ball. 
  Maybe that even helps us to understand the internal
structure of the electron, because we know its magnetic moment and now can try to find out what kind of size and
internal charge distribution goes with that value. Many of the best physicists have tried to do exactly that. 
  However, as it turns out, whatever
assumptions you make about the internal structure of the electron that will give the right magnetic moment will always
get you into deep trouble with other properties
of the electron. There simply is no internal structure of the electron that will
explain its properties! 
  We thus are forced to simply accept as a fundamental property of an
electron that it always carries a magnetic moment of 
 
m^{e} = 
2 · h · e · s 4p · m*_{e}  = ±
m_{Bohr} 


  The factor 2 is a puzzle of sorts  not only because it appears at all, but because it is actually =
2.00231928. But pondering this peculiar fact leads straight to quantum electrodynamics (and several Nobel prizes),
so we will not go into this here. 