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A charged particle of specific charge $\alpha$ is released from origin at time $t = 0$ with velocity $\vec V = {V_o}\hat i + {V_o}\hat j$ in magnetic field $\vec B = {B_o}\hat i$ . The coordinates of the particle at time $t = \frac{\pi }{{{B_o}\alpha }}$ are (specific charge $\alpha = \,q/m$)
$\left( {\frac{{{V_o}}}{{2{B_o}\alpha }},\frac{{\sqrt 2 {V_o}}}{{\alpha {B_o}}},\frac{{ - {V_o}}}{{{B_o}\alpha }}} \right)$
$\left( {\frac{{ - {V_o}}}{{2{B_o}\alpha }},\,0,\,0} \right)$
$\left( {0,\,\frac{{2{V_o}}}{{{B_o}\alpha }},\frac{{{V_o}\pi }}{{2{B_o}\alpha }}} \right)$
$\left( {\frac{{{V_o}\pi }}{{{B_o}\alpha }},\,0,\, - \frac{{2{V_o}}}{{{B_o}\alpha }},} \right)$
Solution
$\alpha=\frac{q}{m},$ path of the particle will be a helix of time period,
$T=\frac{2 \pi m}{B_{0} q}=\frac{2 \pi}{B_{0} \alpha}$
The given time $t=\frac{\pi}{B_{0} \alpha}=\frac{T}{2}$
Coordinates of particle at time $t=T / 2$ would be $\left(v_{x} T / 2,0,-2 r\right)$
Here, $r=\frac{m v_{0}}{B_{0} q}=\frac{v_{0}}{B_{0} \alpha}$
$\therefore$ The coordinate are $\left(\frac{v_{0} \pi}{B_{0} \alpha}, 0, \frac{-2 v_{0}}{B_{0} \alpha}\right)$
