Application of Gauss's Theorem: Field Due to a Uniformly Charged Infinite Plane Sheet

The problem. Find the electric field near an infinite, thin, flat sheet carrying a uniform surface charge density $\sigma$ (Section 1.5).

Exploiting the symmetry. An infinite plane sheet looks identical from every point along its own surface, and it possesses an obvious reflection symmetry: the field at a given perpendicular distance on one side of the sheet must be the mirror image of the field at the same perpendicular distance on the other side. The only field direction consistent with this planar symmetry is one that points exactly perpendicular to the sheet, away from it on both sides (for positive $\sigma$), with a magnitude that depends only on the perpendicular distance from the sheet -- not on position along the sheet, nor (perhaps surprisingly) on the distance from the sheet at all, as the result below will confirm.

Choosing the Gaussian surface. The symmetry is matched by choosing a small "pillbox" -- a short cylinder oriented perpendicular to the sheet, straddling it symmetrically so that it pokes out an equal perpendicular distance on each side, with its two flat circular end caps (each of area $A$) parallel to the sheet, as shown in the figure.

Evaluating the flux, part by part. On each of the two flat end caps, $\vec{E}$ is perpendicular to the sheet and hence exactly parallel to that end cap's own outward normal, with the same magnitude $E$ on both caps (by the reflection symmetry), so each cap contributes flux $EA$; the two caps together give

On the curved side wall of the pillbox (perpendicular to the sheet, running parallel to $\vec{E}$), the field is exactly parallel to this surface and perpendicular to its normal, so it contributes zero flux: $\Phi_{\text{side}}=0$.

Applying Gauss's theorem. The total flux is $\Phi=2EA$. The charge enclosed within the pillbox is just the portion of the sheet lying inside it, of area $A$, carrying charge $q_{\text{enc}}=\sigma A$. Gauss's theorem then gives

directed away from the sheet on both sides for a positive $\sigma$ (toward the sheet on both sides for a negative $\sigma$). …