Question

An electric charge $+q$ moves with velocity $\overrightarrow{V}=3\hat{i}+4\hat{j}+\hat{k}$, in an electromagnetic field given by
$\overrightarrow{E}=3\hat{i}+\hat{j}+2\hat{k}\&\overrightarrow{B}=\hat{i}+\hat{j}+3\hat{k}$.
The $y-$component of the force experienced by $+q$ is :

• A. $-7q$
• B. $10q$
• C. $-11q$
• D. $-9q$

Answer 1

The correct option is A $-7q$

Given:
$\overrightarrow{V}=3\hat{i}+4\hat{j}+\hat{k};\overrightarrow{E}=3\hat{i}+\hat{j}+2\hat{k}$
$\overrightarrow{B}=\hat{i}+\hat{j}+3\hat{k}$.

$\therefore \overrightarrow{V}\times \overrightarrow{B}=(3\hat{i}+4\hat{j}+\hat{k})\times (\hat{i}+\hat{j}+3\hat{k})$

$\overrightarrow{V}\times \overrightarrow{B}=(11\hat{i}-8\hat{j}-\hat{k})$

Electromagnetic force or Lorentz force is given by

$\overrightarrow{F}=q[\overrightarrow{E}+(\overrightarrow{V}\times \overrightarrow{B}\left)\right]$

Putting the value of given parameters, we get

$\overrightarrow{F}=q\left[\right(3\hat{i}+\hat{j}+2\hat{k})+(11\hat{i}-8\hat{j}-\hat{k}\left)\right]$

$\overrightarrow{F}=q(14\hat{i}-7\hat{j}+\hat{k})$

$\therefore F_y=-7q$

Hence, option $\left(a\right)$ is the correct choice.

Why this question ? To familiarize oneself how to use Lorentz force equation in different scenarios and vector calculation familiarity.