Q.An infinite plane sheet carries a uniform surface charge density $\sigma = 8.85\times10^{-8}\ \text{C/m}^2$. Using Gauss's theorem, find the magnitude of the electric field at any point near the sheet (take $\epsilon_0 = 8.85\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}$).

For an infinite, uniformly charged flat sheet (surface density $\sigma$), planar symmetry means $\vec{E}$ is perpendicular to the sheet on both sides, with equal magnitude at equal perpendicular distances. Choosing a small pillbox Gaussian surface straddling the sheet, with flat end-caps of area $A$ on each side, the flux is $2EA$ through the two end-caps (zero through the curved side wall), and the enclosed charge is $\sigma A$; Gauss's theorem gives $E=\sigma/(2\epsilon_0)$, directed away from the sh …