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  
\[\mu=\frac{e \hbar}{2m}\]
  
\[\]
 

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