An electron moving in a circular motion is shown in figure below
Current due to circular motion
$ i = \frac{e}{t}$, where $t = $ time period
So, the magnetic moment of electron
$ m= i.A = \frac{e}{T} .\pi r^{2} $
$ \Rightarrow m = er^{2} \frac{2\pi}{2T} $
$= \frac{e\omega r^{2}}{2} \left[ \because\frac{2\pi}{t} = \omega\right] $
$ \Rightarrow m =\frac{evr}{2} \left[\because \omega= \frac{v}{r}\right]$
