In Bohr model of Hydrogen atom, electron is modeled as a point negative charge rotating in a circular orbit about a fixed axis about a nucleus. Magnetic moment ${\mathrm{M}}_{\mathrm{L}}$
Let us consider
$\mathit{r}=$ radius of the orbit
$\mathit{v}=$ velocity
$\mathit{e}=$ charge of electron
$\mathit{m}=$ mass of electron
Time period
Current(i) $=\frac{-\mathit{e}}{\mathit{T}}=\frac{-\mathit{e}}{\frac{2\mathit{\pi }\mathit{r}}{\mathit{v}}}=\frac{-\mathit{e}\mathit{v}}{2\mathit{\pi }\mathit{r}}$
The magnetic moment due to a current loop enclosing an area $\mathrm{A}$ is given by:
${\mathrm{M}}_{\mathrm{L}}=\mathrm{i}\mathrm{A}=\frac{-\mathit{e}\mathit{v}}{2\mathit{\pi }\mathit{r}}\times \mathrm{A}=\frac{-\mathit{e}\mathit{v}}{2\mathit{\pi }\mathit{r}}\times \mathit{\pi }{\mathrm{r}}^2$ $=\frac{-\mathit{e}\mathit{r}\mathit{v}}{2}=\frac{-\mathit{m}\mathit{e}\mathit{r}\mathit{v}}{2\mathit{m}}$
$\mathrm{L}=$ Angular momentum $=\mathrm{}\mathit{m}\mathit{v}\mathit{r}$
So, ${\mathrm{M}}_{\mathrm{L}}=\frac{-\mathit{e}}{2\mathit{m}}\mathrm{L}$