
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. 