$\mathrm{F}_{\mathrm{E}}=(-\mathrm{e}) \mathrm{E}$

Force on the electron due to the magnetic field $B$ is

$\mathrm{F}_{\mathrm{B}}=(-\mathrm{e}) \mathrm{v} \mathrm{B}$

The electron will move in the fields undeflected, if these two forces are equal and opposite.

$\mathrm{eE}=\mathrm{evB} \quad$ or $\quad \mathrm{v}=\frac{\mathrm{E}}{\mathrm{B}}$

Electric field between the plates is $\mathrm{E}=\frac{\sigma}{\varepsilon_{0}}$

$\therefore \mathrm{v}=\frac{\sigma}{\varepsilon_{0} \mathrm{B}}$

The time taken by the electron to travel a distance $\ell$ in the space is $t=\frac{\ell}{V}=\frac{\ell}{\frac{\sigma}{\varepsilon_{0} B}}=\frac{\ell \varepsilon_{0} B}{\sigma}$