A charged particle going around in a circle can be considered to be a current loop. A particle, of mass m carrying charge q is moving in a plane with speed v under the influence of magnetic field $\overline{\mathrm{B}}.$ The magnetic moment of this moving particle:

Length of the circular path, $\mathrm{\ell }=2\mathrm{\pi r}$

Current, $\dot{\mathrm{i}}=\frac{\mathrm{q}}{\mathrm{T}}=\frac{\mathrm{qv}}{2\mathrm{\pi r}}$ 

Magnetic moment M = Current × Area = $\mathrm{i}\times {\mathrm{\pi r}}^2=\frac{\mathrm{qV}}{2\mathrm{\pi r}}\times {\mathrm{\pi r}}^2$

$\mathrm{M}=\frac{1}{2}\text{ q.v.r }$

Radius of circular path in magnetic field, $\mathrm{r}=\frac{\mathrm{mV}}{\mathrm{qB}}$

$\mathrm{M}=\frac{1}{2}\mathrm{qv}\times \frac{\mathrm{mv}}{\mathrm{qB}}\Rightarrow \mathrm{M}=\frac{{\mathrm{mv}}^2}{2\mathrm{B}}$

Direction of $\overline{\mathrm{M}}$ is opposite of $\overline{\mathrm{B}}$ therefore,

$\overline{\mathrm{M}}=\frac{-{\mathrm{mv}}^2 \overline{\mathrm{B}}}{2{\mathrm{B}}^2}$

(By multiplying both numerator and denominator by B).