question_answer

A particle of mass m and charge q is projected into a uniform magnetic field \[\vec{B}=-{{B}_{0}}\hat{k}\] with velocity \[\overset{\to }{\mathop{v}}\,={{v}_{0}}\hat{i}\] from origin. The position vector of the particle at time t is \[\overset{\to }{\mathop{r}}\,\] . Find the impulse of magnetic force on the particle by the time when \[\overset{\to }{\mathop{r}}\,.\overset{\to }{\mathop{v}}\,\]becomes zero for the first time.

A) \[2m{{v}_{0}}\hat{i}\] 

B)        \[\sqrt{2}m{{v}_{0}}(\hat{i}+\hat{j})\]

C) \[-2m{{v}_{0}}\hat{i}\]

D) \[\sqrt{2}m{{v}_{0}}(\hat{i}-\hat{j})\]

Correct Answer: C

Solution :

[c] The particle goes in a circular path in xy plane. \[\vec{r}.\vec{v}\] when \[\vec{r}{{\bot }_{r}}\vec{V}\] This happens when the particle has completed half circle (see figure) Change in momentum of particle \[\overrightarrow{\Delta p}=-2m{{v}_{0}}\hat{i}\] \[\therefore \]       Impulse  \[=-2m{{v}_{0}}\hat{i}\]