PX157 - B5d - electric field from an infinite plane of charge

σ = \frac{Δ Q}{Δ S}

step 1: symmetry
- $\vec{E}$ points vertically away from the sheet
- $| \vec{E} |$ is constant for a constant distance, $z$ , above or below the sheet
step 2: gaussian surface
- cylinder with axis in the z-direction, intersecting the sheet
$\begin{align*}$ &= \iint E , dS + \iint E , dS + 0 \
&= 2E \iint dS \
&= 2E A
\end{align*}$$
step 3:

Q_{e n c l .} = σ A

\begin{aligned} 2 E A = \frac{σ A}{ϵ_{0}} & ⟹ E = \frac{σ}{2 ϵ_{0}} \\ ∴ \vec{E} & = {\begin{cases} \frac{σ}{2 ϵ_{0}} \hat{z} & z > 0 \\ - \frac{σ}{2 ϵ_{0}} & z < 0 \end{cases} \end{aligned}