Gauss's Law

Gauss’s law says that if you imagine any closed surface around some charges, the net electric flux through that surface is set only by the total charge inside it, not by the surface’s size or shape. It works because electric field lines originate on positive charge and terminate on negative charge, so the net number crossing a closed boundary is a direct measure of enclosed charge. In symmetric situations, this turns a difficult field problem into a much simpler flux calculation.

Use this law when a problem has high symmetry—spherical, cylindrical, or planar—because it can let you find the electric field much faster than summing contributions from individual charges. It is also useful for checking whether a proposed electric field is physically consistent with a given charge distribution.

• using the total charge outside the surface instead of only the enclosed charge
• forgetting that Gauss’s law gives flux through a closed surface, not the electric field directly
• missing the sign of the enclosed charge, which determines whether the net flux is positive or negative