Question

In a certain region of space, electric field $\overrightarrow{E}$ and magnetic field $\overrightarrow{B}$ are perpendicular to each other. An electron enters in the region perpendicular to the directions of both $\overrightarrow{B}$ and $\overrightarrow{E}$ and moves undeflected. Find the velocity of the electron.

Answer 1

Net force on electron moving in the combined electric field $\overrightarrow{E}$ and magnetic field $\overrightarrow{B}$ is
$\overrightarrow{F}=\overrightarrow{E}+(\overrightarrow{v}\times \overrightarrow{B})$
Since electron moves undeflected then $\overrightarrow{F}=0$.
$\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}=0$
$\Rightarrow |\overrightarrow{E}|=|\overrightarrow{v}|\times |\overrightarrow{B}|\to |\overrightarrow{v}|=\frac{|\overrightarrow{E}|}{|\overrightarrow{B}|}$