A thin infinite sheet charge and an infinite line charge of respective charge densities $+\sigma$ and $+\lambda$ are placed parallel at $5\ \text{m}$ distance from each other. Points $P$ and $Q$ are at $\dfrac3\pi$ m and $\dfrac4\pi$ m perpendicular distances from line charge towards sheet charge, respectively. $E_P$ and $E_Q$ are the magnitudes of resultant electric field intensities at point $P$ and $Q$ respectively. If $\dfrac{E_P}{E_Q}=\dfrac4a$ for $2|\sigma|=|\lambda|$, then the value of $a$ is ______.
Numerical value type. Enter your answer.
Answer: 6
Between them the two fields oppose: the line pushes away from itself (towards the sheet) with $\dfrac{\lambda}{2\pi\epsilon_0r}$, the sheet pushes away from itself with $\dfrac{\sigma}{2\epsilon_0}$. With $\lambda=2\sigma$:
$$E_P=\left|\frac{2\sigma}{2\pi\epsilon_0(3/\pi)}-\frac{\sigma}{2\epsilon_0}\right|=\frac{\sigma}{\epsilon_0}\left|\frac13-\frac12\right|=\frac{\sigma}{6\epsilon_0}$$
$$E_Q=\frac{\sigma}{\epsilon_0}\left|\frac14-\frac12\right|=\frac{\sigma}{4\epsilon_0}$$
$\dfrac{E_P}{E_Q}=\dfrac23=\dfrac46$, so $a=6$.
Solution by Sreeraj P, M.Sc Physics