$A$ charged particle of charge $e$ and mass $m$ is moving in an electric field $\vec{E}$ and magnetic field $\vec{B}$. Construct dimensionless quantities and quantities of dimension $T^{-1}$.

(A) The Lorentz force on a charged particle is given by $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$.
For a particle moving in a magnetic field $\vec{B}$ with velocity $\vec{v}$ perpendicular to the field,the magnetic force provides the centripetal force:
$qvB = \frac{mv^2}{R}$
Rearranging this,we get the cyclotron frequency $\omega = \frac{v}{R} = \frac{qB}{m}$.
The dimensions of $\omega$ are:
$[\omega] = \frac{[q][B]}{[m]} = \frac{[I][T][M][I]^{-1}[T]^{-2}}{[M]} = [T]^{-1}$.
Thus,the quantity $\frac{qB}{m}$ has the dimension $T^{-1}$.
Additionally,the ratio of the electric field to the magnetic field $\frac{E}{B}$ has the dimension of velocity $[L][T]^{-1}$,and the quantity $\frac{eE}{mv}$ or $\frac{eB}{m}$ can be used to construct various dimensionless ratios depending on the physical context.