3.2.7 Problems and solutions

Every problem in this subsection is solved with Gauss's law in integral form. In free space ($D={\epsilon }_{0}E$) we write it as

The left-hand side is evaluated on a Gaussian surface chosen so that, on each part of it, $E$ is either parallel to $dS$ with constant magnitude or perpendicular to it. The right-hand side is the charge enclosed. Equating both sides gives $|E|$. The fluxes that occur again and again are $2\pi \rho L{\epsilon }_0{E}_{\rho }$ on a coaxial cylinder of radius $\rho$ and length $L$, $4\pi r^2{\epsilon }_0{E}_r$ on a concentric sphere of radius $r$, and $2A{\epsilon }_0{E}_z$ through a pillbox of face area $A$ across a plane. In the plots, the shaded bands mark the regions of the problem (charged material in colour, empty space white).