Three infinitely long, thin charged sheets are placed parallel to the $y$-$z$ plane at $x = -a$ (surface charge density $-\sigma$), $x = a$ (density $-2\sigma$) and $x = 3a$ (density $+\sigma$). The magnitude of the electric field at a point $P$ lying between $x = a$ and $x = 3a$ is $\dfrac{x\sigma}{\varepsilon_0}$. The value of $x$ is ______ (all quantities are measured in SI units).
Numerical value type. Enter your answer.
Answer: 2
Each sheet gives $\dfrac{|\sigma'|}{2\varepsilon_0}$, directed away from a positive sheet and towards a negative one. At $P$ (between $x = a$ and $x = 3a$):
- $-\sigma$ at $x = -a$ (to the left): towards it, $-x$ direction, $\dfrac{\sigma}{2\varepsilon_0}$
- $-2\sigma$ at $x = a$ (to the left): towards it, $-x$ direction, $\dfrac{\sigma}{\varepsilon_0}$
- $+\sigma$ at $x = 3a$ (to the right): away from it, $-x$ direction, $\dfrac{\sigma}{2\varepsilon_0}$
$$E = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{2\sigma}{\varepsilon_0} \Rightarrow x = 2$$
Solution by Sreeraj P, M.Sc Physics