
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. 