Uniform electric and magnetic fields with strength $E$ and $B$ are directed along the $y$-axis. A particle with specific charge $\raisebox{1ex}{$q$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$ leaves the origin in the direction of x-axis with an initial velocity $v_0$. Find the y-coordinate of the particle when it crosses the y-axis for n^th time.

The path of the particle is a helix of increasing pitch. The axis of the helix is parallel to y-axis (parallel to $E$) and plane of circle of the helix is (perpendicular to $B$). The particel will cross the y-axis after time,

$t=nT=n\left(\frac{2\mathrm{\pi m}}{Bq}\right)=\frac{2\mathrm{\pi mn}}{Bq}$

The y-coordinate of the particle at this instant is $y=\frac{1}{2}{a}_y{t}^2$,

where, $a_y=\frac{F_y}{m}=\frac{qE}{m}$

$$\begin{gathered} \therefore y=\frac{1}{2}\left(\frac{qE}{m}\right){\left(\frac{2\mathrm{\pi mn}}{Bq}\right)}^2 \\ =\frac{2n^{2}mE{\mathrm{\pi }}^2}{q{B}^2} \end{gathered}$$