A particle of charge q and mass m starts moving from the origin under the action of an electric field and $\overrightarrow{E}={\text{E}}_{\text{0}}\hat{i}\text{and}\overrightarrow{B}={\text{B}}_{\text{0}}\hat{i}$ with velocity $\overrightarrow{v}={\text{v}}_{\text{0}}\hat{j}$ . The speed of the particle will become $2{\text{v}}_{\text{0}}$  after a time

Acceleration along x – axis due to electric field

${\text{a}}_x=\frac{\text{Eq}}{\text{m}}$

Velocity of particle after Time “t” along x – axis

${\text{V}}_x=\text{u}+\text{at}$

${\text{V}}_x=0+\frac{\text{Eq}}{\text{m}}\text{t}=\frac{\text{Eq}}{\text{m}}\text{t}$  

(Initial velocity along x – axis = 0)

Due to magnetic field particle deflects into Y – Z plane

${\text{V}}_y={\text{V}}_{\text{0}}\cos \theta$

${\text{V}}_x={\text{V}}_{\text{0}}\sin \theta$

$\overrightarrow{V}={\text{V}}_x\hat{i}+{\text{V}}_y\hat{j}+{\text{V}}_x\hat{k}$

$2{\text{V}}_0=\sqrt{{\text{V}}_x^2+{\text{V}}_y^2+{\text{V}}_z^2}$

$2{\text{V}}_0=\sqrt{{\left(\frac{\text{Eq}}{\text{m}}\text{t}\right)}^2+{\text{V}}_0^2{\cos }^2\theta +{\text{V}}_0^2{\sin }^2\theta }$

$4{\text{V}}_0^2={\left(\frac{\text{Eq}}{\text{m}}\text{t}\right)}^2+{\text{V}}_0^2$

$\text{t}=\frac{\sqrt{3}{\text{mV}}_0}{\text{Eq}}$