In a circular motion of charge particle, the charge particle moving perpendicular to the magnetic field, the magnetic Lorentz forces provides necessary centripetal force for revolution.
Contripental force $\left(\frac{\mathrm{mv}^2}{\mathrm{R}}\right)$ is balanced by magnetic force $\left(\mathrm{F}_{\mathrm{m}}\right)=\mathrm{qvB} \sin 90^{\circ}=\mathrm{qvB}$ then, $\frac{m v^2}{R}=q v B$
By solving, we get
So, $\frac{\mathrm{qB}}{\mathrm{m}}=\frac{\mathrm{v}}{\mathrm{R}}=\omega \quad(\therefore \mathrm{V}=\omega \mathrm{R}$ and $\mathrm{q}=\mathrm{e})$
Then the dimensional formula of angular frequency is $\therefore \quad[\omega]=\left[\frac{\mathrm{eB}}{\mathrm{m}}\right]=\left[\frac{\mathrm{V}}{\mathrm{R}}\right]=[\mathrm{T}]^{-1}$
that is the required expression.