The magnetic moment of an electron (e) revolving in an orbit around nucleus with an orbital angular momentum is given by :

The current i due to the motion of the electron is the charge flowing through that time period,

$\mathrm{i}=\frac{-\mathrm{e}}{\frac{2\mathrm{\pi r}}{\mathrm{v}}}=\frac{-\mathrm{ev}}{2\mathrm{\pi r}}$

$\overrightarrow{\mathrm{\mu }}=\frac{-\mathrm{ev}}{2\mathrm{\pi r}}\mathrm{A}=\frac{-\mathrm{ev}}{2\mathrm{\pi r}}{\mathrm{\pi r}}^2$...(i)

$\Rightarrow \overrightarrow{\mathrm{\mu }}=\frac{-\mathrm{e}}{2{\mathrm{m}}_{\mathrm{e}}}{\mathrm{m}}_{\mathrm{e}}\mathrm{vr}$

we know, for electron  $\overrightarrow{\mathrm{L}}={\mathrm{m}}_{\mathrm{e}}\mathrm{vr}$

$\Rightarrow \overrightarrow{\mathrm{\mu }}=-\frac{\mathrm{e}}{2{\mathrm{m}}_{\mathrm{e}}}\overrightarrow{\mathrm{L}}$