A particle of specific charge $(q/m)$ is projected from the origin of coordinates with initial velocity $[ui - vj]$. 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
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Taking motion along $y$ axis (con by electric field)
$\alpha=y_{0}+u_{y} t+\frac{1}{2} q_{y} t^{2}$
$0=0-v t+\frac{1}{2} \frac{q E}{m} t^{2}$
$t=\frac{2 m v}{q E}$
In this time charge must complex one oe more revolution in $\mathrm{x}$$-$ $\mathrm{z}$ plane due to magnetic field
$T=\frac{2 \pi m}{q B}=-2 \Rightarrow t=n t \ldots(2)$
$\frac{2 m v}{q E}=n \times \frac{2 \pi m}{q B} \Rightarrow \frac{v B}{\pi E}$
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