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There exists a uniform magnetic and electric field of magnitude $1\, T$ and $1\, V/m$ respectively along positive $y-$ axis. A charged particle of mass $1\,kg$ and of charge $1\, C$ is having velocity $1\, m/sec$ along $x-$ axis and is at origin at $t = 0.$ Then the co-ordinates of particle at time $\pi$ seconds will be :-
$(0,1,2)$
$(0, - {\pi ^2}/2, - 2)$
$(2, {\pi ^2}/2, 2)$
$(0, {\pi ^2}/2, 2)$
Solution
The particle will move in a non-uniform helical path with increasing pitch as shown below:
Its time period will be:
Changing the view, the particle is seemed to move in a circular path in $(\mathrm{x}-\mathrm{z})$ plane as below
Its time period will be $\mathrm{T}=\frac{2 \pi \mathrm{m}}{\mathrm{qB}}=2 \pi \mathrm{sec}$
Changing the view the particle is seemed to move in a circular path in $(\mathrm{x}-\mathrm{z})$ plane as below
After $\pi$ -seconds the particle will be at point $'P',$ hence $x$ coordinate will be $0$
For linear motion along $y$ – direction.
$\mathrm{y}(\pi)=0(\pi)+\frac{1}{2} \frac{\mathrm{E} \mathrm{q}}{\mathrm{m}}(\pi)^{2}$
$\mathrm{y}(\pi)=\frac{\pi^{2}}{2}$ and $\mathrm{OP}=2$ Hence the coordinate
$\left(0, \frac{\pi^{2}}{2}, 2\right)$
