An infinite plane sheet of charge having uniform surface charge density $+\sigma_s\ \text{C/m}^2$ is placed on the $x$-$y$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e\ \text{C/m}$ is placed in the $z = 4\ \text{m}$ plane and parallel to the $y$-axis. If the magnitudes $|\sigma_s| = 2|\lambda_e|$, then at the point $(0, 0, 2)$ the ratio of the magnitudes of the electric field due to the sheet charge to that due to the line charge is $\pi\sqrt n : 1$. The value of $n$ is ______.
Numerical value type. Enter your answer.
Answer: 16
Sheet: $E_s = \dfrac{\sigma_s}{2\varepsilon_0}$.
The line (taken through $x = 0$, $z = 4$) is $2\ \text{m}$ from the point: $E_l = \dfrac{\lambda_e}{2\pi\varepsilon_0(2)} = \dfrac{\lambda_e}{4\pi\varepsilon_0}$.
$$\frac{E_s}{E_l} = \frac{\sigma_s/2\varepsilon_0}{\lambda_e/4\pi\varepsilon_0} = \frac{2\pi\sigma_s}{\lambda_e} = 4\pi = \pi\sqrt{16}$$
So $n = 16$.
Solution by Sreeraj P, M.Sc Physics