Question

A particle of charge $q$ and mass $m$ starts moving from the origin with a velocity $\vec{v}=v_{0} \hat{j}$ under the action of an electric field $\vec{E}=E_{0} \hat{i}$ and magnetic field $\vec{B}=B_{0} \hat{i}$. The speed of the particle will become $2 v_{0}$ after a time $t=\frac{\sqrt{x} \cdot m v_{0}}{q E_{0}} .$ Find the value of $x$.

Answer 1

Let the velocity of particle along $x$-axis be $v_{x}$.
$\Rightarrow \sqrt{v_{0}^{2}+v_{x}^{2}}=2 v_{0} $
$v_{x}=\sqrt{3} v_{0} $
$a_{x}=\frac{q E_{0}}{m} $
$\Rightarrow v_{x}=a_{x} t $
$\Rightarrow \sqrt{3} v_{0}=\frac{q E_{0} t}{m}$
$\Rightarrow t=\frac{\sqrt{3} v_{0} m}{q E_{0}}$