Question

A charged particle goes undeflected in a region of space containing an electric field $\overrightarrow{E}$ and a magnetic field of intensity $\overrightarrow{B}$. Which of the following is possible?

• A. $\overrightarrow{E}$ is parallel to $\overrightarrow{B}$ but $\overrightarrow{v}$ is perpendicular to $\overrightarrow{E}$
• B. $\overrightarrow{E}$ is parallel to $\overrightarrow{B}$ and $\overrightarrow{v}$ parallel to $\overrightarrow{E}$
• C. $\overrightarrow{E}$ is perpendicular to $\overrightarrow{B}$ and $\overrightarrow{v}$ is parallel to $\overrightarrow{E}\times \overrightarrow{B}$
• D. $\overrightarrow{E}$ is perpendicualar to $\overrightarrow{B}$ and $\overrightarrow{v}$ is parallel to $\overrightarrow{E}.\overrightarrow{B}$

Answer 1

Answer: (B), (C)

$\overrightarrow{E}$ is perpendicular to $\overrightarrow{B}$ and $\overrightarrow{v}$ is parallel to $\overrightarrow{E}\times \overrightarrow{B}$

From Lorentz force,
$\overrightarrow{F_L}=\overrightarrow{F_e}+\overrightarrow{F_m}$
$\overrightarrow{F_e}$ = electric force
$\overrightarrow{F_m}$ = magnetic force
$\overrightarrow{F_L}=q\overrightarrow{E}+q(\overrightarrow{v}\times \overrightarrow{B})$

If $\overrightarrow{E}||\overrightarrow{B}$ and $\overrightarrow{v}||\overrightarrow{E}$, then $\overrightarrow{v}$ must be parallel to $\overrightarrow{B}.$
$\therefore \overrightarrow{F_L}=q\overrightarrow{E}+0=q\overrightarrow{E}$
Charge will either accelerate or decelerate in the same direction of $\overrightarrow{v},$ but does not deflect.

If $\overrightarrow{E}$ is perpendicular to $\overrightarrow{B}$ and $\overrightarrow{v}$ is parallel to $\overrightarrow{E}\times \overrightarrow{B}$
$\Rightarrow$ $\overrightarrow{E},$ $\overrightarrow{B}$ and $\overrightarrow{v}$ are mutually perpendicular to each other.
$\Rightarrow \overrightarrow{v}\times \overrightarrow{B}=k\overrightarrow{E}$

$\overrightarrow{F_L}=q\overrightarrow{E}+q(\overrightarrow{v}\times \overrightarrow{B})$
$\overrightarrow{F_L}=q\overrightarrow{E}+qk\overrightarrow{E}$
For $k=-1,F_L=0$
Hence, its possible for charge to travel undeflected.