A large flat metal surface has a uniform charge density $\left(+\sigma \right)$ . An electron of mass $m$ and charge $\rho$ leaves the surface at the point $\mathrm{A}$ with speed $u$ and returns to it at point $\mathrm{B}$. Disregarding gravity the maximum value of $\mathrm{AB}$ is

• A

  $\frac{{\mathrm{4u}}^{2}m{\epsilon }_0}{\sigma \rho }$
• B

  $\frac{u^2\cdot \rho {\epsilon }_0}{m\sigma }$
• Correct answer
  C

  $\frac{2{\epsilon }_{0}m{u}^2}{\rho \sigma }$
• D

  $\frac{u^2\sigma \rho }{\epsilon \cdot m}$

Solution

Electric field due to one surface of the large metal plate,



The projected electron's motion will be on a curved path like a projectile.

$\therefore$ AB will be maximum when $\theta =45°$

Range $A{B}_{max}=\frac{u^2}{a}$

$a=\frac{\rho E}{m}=\frac{\mathrm{\rho }\sigma }{2{\epsilon }_{0}m}$

$=\frac{u^2}{\frac{\rho \sigma }{2{\epsilon }_{0}m}}=\frac{2{\epsilon }_{0}m{u}^2}{\rho \sigma }$

Hence, the correct answer is option: 'C'.