question_answer

The region between\[y=0\]and\[y=d\]contains a magnetic field \[\vec{B}=B\overset{\wedge }{\mathop{z}}\,.\] A particle of mass m and charge q enters the region with a velocity \[\vec{v}=v\hat{i}.\] If \[d=\frac{mv}{2qB},\] the acceleration of the charged particle at the point of its emergence at the other side is [JEE  Main Online Paper (Held on 11-jan-2019 Evening)]

A) \[\frac{qvB}{m}\left( \frac{\hat{i}+\hat{j}}{\sqrt{2}} \right)\]                              

B) \[\frac{qvB}{m}\left( \frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j} \right)\]

C) \[\frac{qvB}{m}\left( \frac{1}{2}\hat{i}-\frac{\sqrt{3}}{2}\hat{j} \right)\]  

D)   \[\frac{qvB}{m}\left( \frac{-\hat{j}+\hat{i}}{\sqrt{2}} \right)\]

E)   None of these

Correct Answer: E

Solution :

\[d=\frac{mv}{2qB}\] As\[r=\frac{mv}{qB}=2d\] Acceleration, \[{{\vec{v}}_{1}}={{v}_{1}}\sin {{30}^{o}}\hat{i}+{{v}_{1}}\cos {{30}^{o}}\hat{j}\] \[=\frac{v}{2}\hat{i}-\frac{\sqrt{3}}{2}v\hat{j}\] \[\vec{a}=\frac{{\vec{F}}}{m}=\frac{qvB}{2m}(\hat{i}-\sqrt{3}\hat{j})\] *None of the given options is correct.