An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet $S$ having surface charge density $+\sigma$. The electron at $t = 0$ is at a distance of $1\ \text{m}$ from $S$ and has a speed of $1\ \text{m s}^{-1}$. The maximum value of $\sigma$, if the electron strikes $S$ at $t = 1\ \text{s}$, is $\alpha\left[\dfrac{m\epsilon_0}{e}\right]\dfrac{\text{C}}{\text{m}^2}$. The value of $\alpha$ is ______.
Numerical value type. Enter your answer.
Answer: 8
The field of the sheet is $\dfrac{\sigma}{2\epsilon_0}$, so the electron has a constant acceleration $a = \dfrac{e\sigma}{2\epsilon_0 m}$ towards the sheet.
The largest $\sigma$ (largest $a$) is needed when the electron initially moves away from the sheet. Taking the distance from the sheet as $x$:
$$x = 1 + (1)t - \frac12at^2 = 0\ \text{at}\ t = 1\ \text{s} \;\Rightarrow\; a = 4\ \text{m s}^{-2}$$
$$\frac{e\sigma}{2\epsilon_0 m} = 4 \;\Rightarrow\; \sigma = 8\,\frac{m\epsilon_0}{e}$$
So $\alpha = 8$.
Solution by Sreeraj P, M.Sc Physics