## Atomic Magnetic Moments

Moving electric charges constitute electric currents, and generate magnetic fields. We can consider an electric circling an atomic nucleus to be a tiny electric current generating a tiny magnetic field. The orbital period is
$T= \frac{2 \pi r}{v}$
s. The current will pass a certain point on its orbit many times
$n=\frac{v}{2 \pi r}$
per second. The current of an electron in an atomic orbit is then
$I=ne= \frac{ev}{2 \pi r}$
.
The orbital angular momentum of the electron is
$L=mvr \rightarrow vr=\frac{L}{2m}$
.
we can define the magnetic moment of the electron in orbit as
$\mu =IA$
where
$A=\pi r^2$
is the area enclosed by the orbit. Using this and the above expression for
$I$
give
$\mu = IA= \frac{ev}{2mr} \pi r^2 = \frac{evr}{2} = \frac{eL}{2m}$
.
Angular momentum in atoms is quantized in units of
$\hbar = \frac{h}{2 \pi}$
so
$\mu=\frac{e \hbar}{2m}$