3.2 Gauss's Law

In Section 3.1 we found the electric field of a charge distribution by cutting it into small pieces, writing Coulomb's law for each piece and adding up the vectors. That always works, but it is laborious: for a uniformly charged sphere, for instance, the vector integral over the volume is a real effort, even though the answer turns out to be very simple.

In this section we look at the same physics from a different angle. Instead of asking "what is the field at one point?", we ask "how much field comes out of a closed surface?". Think of a lawn sprinkler hidden inside a large tent: you cannot see the sprinkler, but by measuring how much water comes out through the walls of the tent you can tell how strong the sprinkler is, wherever it stands inside the tent and whatever the shape of the tent. Gauss's law is exactly this statement for electric charges: the field coming out of a closed surface counts the charge inside it.

The result is one of the four Maxwell equations. As a calculating tool it gives the field of every sufficiently symmetric source in a few lines. As a reasoning tool it tells us where the charge on a conductor sits and why a closed metal box shields its interior.