Magnetic field $\overrightarrow{B}=-{\text{B}}_{\text{0}}x\hat{k}$ exists in a region of space. A particle of specific charge $\alpha$  (charge per unit mass) enters this region of space. Its velocity and position at time $\text{t}=0,\text{are}\overrightarrow{v}={\text{v}}_{\text{0}}\hat{i}$ and (0,0,0). Find the maximum x-coordinate of the particle.

$\overrightarrow{B}=-{\text{B}}_{\text{0}}x\hat{k}\text{at}x=0$

$\Rightarrow \overrightarrow{B}=0\text{and}\overrightarrow{V}={\text{V}}_{\text{0}}\hat{i}$

Magnetic force F = 0 at the origin as particle moves magnetic force in x – y plane

$\therefore$ Total motion is confined into x – y plane

When ${\text{V}}_y={\text{V}}_{\text{0}}$ and${\text{V}}_x=0$ then

$x=\max$

$\overrightarrow{F}=q\left(\overrightarrow{V}\times \overrightarrow{B}\right)=\text{q}\left({\text{V}}_x\hat{i}+{\text{V}}_y\hat{j}\right)\times \left(-{\text{B}}_{\text{0}}x\hat{k}\right)$

$\text{m}\overrightarrow{a}=\text{q}\left({\text{B}}_{\text{0}}x{\text{V}}_x\hat{j}-{\text{B}}_{\text{0}}x{\text{V}}_y\hat{i}\right)$

$\overrightarrow{a}=\frac{\text{V}}{\text{m}}\left({\text{B}}_{\text{0}}x{\text{V}}_x\hat{j}-{\text{B}}_{\text{0}}x{\text{V}}_y\hat{i}\right)$

${\text{a}}_{\text{y}}={\text{B}}_{\text{0}}\alpha x{\text{V}}_x$

$\frac{{\text{dV}}_y}{\text{d}x}.\frac{\text{d}x}{\text{dt}}={\text{B}}_{\text{0}}\alpha x{\text{V}}_x$

${\text{dV}}_{\text{y}}{\text{.V}}_x={\text{B}}_{\text{0}}\alpha x{\text{V}}_x.\text{d}x$

${\text{V}}_{\text{y}}={\text{B}}_{\text{0}}\alpha x{\text{V}}_x.\text{d}x$

${\text{V}}_{\text{0}}=\frac{{\text{B}}_{\text{0}}\alpha }{2}{x}^2$

$\Rightarrow x=\sqrt{\frac{2V_0}{B_0\alpha }}$