question_answer

A charged particle of specific charge (charge/mass) \[\alpha \] is projected from origin with a velocity \[\vec{u}={{v}_{0}}\,(\hat{i}+\hat{j})\] in a uniform and constant magnetic field \[\vec{B}={{B}_{0}}\hat{i}.\] The position co-ordinates of the particle at time \[t=\frac{\pi }{{{B}_{0}}\alpha }\] are

A) \[\left( \frac{{{v}_{0}}}{2{{B}_{0}}\alpha },\,\,\frac{\sqrt{2}{{v}_{0}}}{\alpha {{B}_{0}}},\,\,\frac{-{{v}_{0}}}{{{B}_{0}}\alpha } \right)\]

B) \[\left( -\frac{{{v}_{0}}}{2{{B}_{0}}\alpha },\,\,0,\,\,0 \right)\]

C) \[\left( 0,\,\,\frac{2{{v}_{0}}}{{{B}_{0}}\alpha },\,\,\frac{{{v}_{0}}\pi }{2{{B}_{0}}\alpha } \right)\]

D) \[\left( \frac{{{v}_{0}}\pi }{{{B}_{0}}\alpha },\,\,0,\,\,\frac{-\,2{{v}_{0}}}{{{B}_{0}}\alpha } \right)\]

Correct Answer: D

Solution :

Radius of projection of helix will be \[r=\frac{{{v}_{0}}}{\alpha {{B}_{0}}}\] and time period of projection will be \[T=\frac{2\pi }{\alpha {{B}_{0}}},\] projected circle will be formed on y-z plane. It will make half circle in time\[t=\frac{\pi }{{{B}_{0}}\alpha }.\] x-coordinate \[={{\upsilon }_{0}}t={{\upsilon }_{0}}\cdot \frac{\pi }{{{B}_{0}}\alpha }.\]