A particle of specific charge (q/m) is projected from the origin of coordinates with initial velocity $[\mathrm{u}\widehat{\mathrm{i}}-\mathrm{v}\widehat{\mathrm{j}}]$. Uniform electric magnetic fields exist in the region along the +y direction, of magnitude E and B. The particle will definitely return to the origin once if

${\overline{\mathrm{B}}}_0=\mathrm{B}\widehat{\mathrm{j}};{\mathrm{E}}_0=\mathrm{E}\widehat{\mathrm{j}};\text{ specific charge }\frac{\mathrm{q}}{\mathrm{m}}$

$\text{ from }\mathrm{S}=\mathrm{ut}+\frac{1}{2}{\mathrm{at}}^2\Rightarrow \mathrm{O}=-{\mathrm{V}}_0\mathrm{t}+\frac{1}{2}\left(\frac{\mathrm{Eq}}{\mathrm{m}}\right){\mathrm{t}}^2$

$\mathrm{t}=\frac{2{\mathrm{V}}_0\mathrm{m}}{\mathrm{Eq}}$ is the time taken by the particle to came to origin due to electric field.
At the same time due to magnetic field along z-axis $\mathrm{d}=\mathrm{R}-\mathrm{Rcos}\mathrm{\theta }\Rightarrow \mathrm{O}=\mathrm{R}(1-\cos \mathrm{\omega t})$ $\Rightarrow 1-\cos \left(\frac{\mathrm{Bq}}{\mathrm{m}}\mathrm{t}\right)=0\Rightarrow \cos \left(\frac{\mathrm{Bq}}{\mathrm{m}}\frac{2{\mathrm{V}}_0\mathrm{m}}{\mathrm{Eq}}\right)=1$

$\frac{2{\mathrm{BV}}_0}{\mathrm{E}}=2\mathrm{n\pi }\Rightarrow \frac{{\mathrm{BV}}_0}{\mathrm{\pi E}}=\mathrm{n}$. It is an integer