3.2.6 Recap

Flux density: $D={\epsilon }_{0}E$ $[C/{\mathrm{m}}^2]$ in free space. For a point charge $D=q{a}_r/(4\pi r^2)$, the charge divided by the area of the sphere through P.

Flux: $\Psi ={\int }_{S}D\cdot dS$ $[\mathrm{C}]$, "how much $D$ passes through". Only the normal component counts, $\Psi =DS\cos \theta$ for a uniform field through a flat surface. Closed surfaces: outward normal, out $=+$, in $=-$.

Point charge: ${\oint }_{S}D\cdot dS=q$ if $q$ is inside $S$, $0$ if outside, Eq. (3.2.8), for any shape of $S$ (the field lines of $q$ cross $S$ and the sphere alike, which works because of the $1/r^2$ of Coulomb's law).

Gauss's law, Eq. (3.2.12):

linked by the divergence theorem. This is the second fundamental equation of electrostatics. Only the enclosed charge counts.

Using it: symmetry $\Rightarrow$ direction and dependence of $D$ $\Rightarrow$ Gaussian surface on which $D$ is normal with constant magnitude, or tangential: pillbox (flux $2DA$), coaxial cylinder ($D2\pi \rho L$), concentric sphere ($D4\pi r^2$). The law always holds, but it is only useful with symmetry.

Standard results: line $E={\rho }_L/(2\pi {\epsilon }_0\rho )$, sheet $E={\rho }_S/(2{\epsilon }_0)$, just outside a conductor $E_n={\rho }_S/{\epsilon }_0$, outside any spherical charge $E=Q/(4\pi {\epsilon }_0{r}^2)$, and inside a uniform sphere $E={\rho }_{v}r/(3{\epsilon }_0)$.

Conductors in equilibrium: $E=0$ inside, net charge on the surface, $D_n={\rho }_S$ just outside. Hence the Faraday cage and the confinement of the field in a coaxial cable.