We can picture an electron in an atom moving in a circle of radius
with speed
The moving electron is equivalent to a current loop. A current loop with area
and current
has a magnetic dipole moment given by
so for the electron above
To find the current associated with the electron, notice that the time for one orbit is
so the current associated with the electron,
The magnetic moment
is then
We can express
in terms of the angular momentum
of the electron.
For electrons in atoms, angular momentum is quantized. It's component in a particular direction is an integral multiple of
If
then
is called the Bohr magneton and denoted
It's numerical value is