Question

A positive charge particle with velocity $\overrightarrow{v}=x\hat{i}+y\hat{j}$ moves in a magnetic field $\overrightarrow{B}=y\hat{i}+x\hat{j}$. The magnitude of magnetic force acting on the particle is F. Which one of the following statements are correct ?

• A. no force will act on particle if $x=y$
• B. $F\propto (x^2-y^2)$ if $x>y$
• C. the force will act along positive Z-axis if $x>y$
• D. the force will act along positive Z-axis if $y>x$

Answer 1

Answer: (A), (B), (C)

The correct options are
A no force will act on particle if $x=y$
B $F\propto (x^2-y^2)$ if $x>y$
C the force will act along positive Z-axis if $x>y$

The magnetic force on a particle moving with velocity '$v$' in a given magnetic field is given by,
$\overrightarrow{F}=\overrightarrow{v}\times \overrightarrow{B}$
$\overrightarrow{F}=(x\hat{i}+y\hat{j})\times (y\hat{i}+x\hat{j})$
$\overrightarrow{F}=x^2\hat{k}+y^2(-\hat{k})$
$\overrightarrow{F}=x^2\hat{k}-y^2\hat{k}$
If $x=y,$ then $F=(x^2-y^2)\hat{k}=0$
$F>0ifx>y$.
So the force acts in positive $z-$ direction if $x>y$