Question

A large flat metal surface has uniform charge density $+\sigma$. An electron of mass $m$ and charge $e$ leaves the surface at an angle from a point $A$ with speed $v$, and returns to it at point $B$. The maximum value of $AB$ is

• A. $\frac{vm{ϵ}_0}{\sigma e}$
• B. $\frac{v^{2}m{ϵ}_o}{\sigma e}$
• C. $\frac{v^{2}e}{{ϵ}_o\sigma m}$
• D. $\frac{v^2\sigma e}{{ϵ}_{o}m}$

Answer 1

The correct option is B $\frac{v^{2}m{ϵ}_o}{\sigma e}$

Electric Field due to metal surface near it $\overrightarrow{E}=\frac{\sigma }{{ϵ}_o}$.

The particle will travel along a parabolic path as $\overrightarrow{E}$ is perpendicular to the surface.

Now $AB$ is range of the particle , $AB$ is maximum when range is maximum

Maximum range of a projectile projected with velocity $v$ is
$R_{max}=\frac{v^2}{a}$
where $ma=e\overrightarrow{E}=\frac{e\sigma }{{ϵ}_o}$
$\Rightarrow a=\frac{e\sigma }{m{E}_o}$


$\therefore R_{max}=AB=\frac{v^{2}m}{e\sigma }{ϵ}_o$