A particle of mass $m$ and charge q has an initial velocity


A particle of mass $m$ and charge $q$ has an initial velocity

$\vec{v}=v_{0} \hat{j}$. If an electric field $\vec{E}=E_{0} \vec{i}$ and magnetic field

$\vec{B}=B_{0} \hat{i}$ act on the particle, its speed will double after a time:

  1. (1) $\frac{2 m v_{0}}{q E_{0}}$

  2. (2) $\frac{3 m v_{0}}{q E_{0}}$

  3. (3) $\frac{\sqrt{3} m v_{0}}{q E_{0}}$

  4. (4) $\frac{\sqrt{2} m v_{0}}{q E_{0}}$

Correct Option: , 3


(3) In the $x$ direction

$F_{x}=q E$

$\Rightarrow \quad m a_{x}=q E$

$\Rightarrow \quad a_{x}=\frac{E_{0} q}{m}$

For speed to be double,

$v_{0}^{2}+v_{x}^{2}=\left(2 v_{0}\right)^{2}$

$\Rightarrow v_{x}=\sqrt{3} v_{0}=a_{x} t$

$\Rightarrow \sqrt{3} v_{0}=0+\frac{q E_{0} t}{m} \Rightarrow t=\frac{\sqrt{3} v_{0} m}{E_{0} q}$

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