In one sense, Gauss’s law doesn’t say anything new or anything that we didn’t already know from Coulomb’s law. After all, we derived Gauss’s law from Coulomb’s law. But in another sense, Gauss’s law is more important than Coulomb’s law. Gauss’s law states a very general property of electric fields—namely, that charges create electric fields in just such a way that the net flux of the field is the same through any surface surrounding the charges, no matter what its size and shape may be. This fact may have been implied by Coulomb’s law, but it was by no means obvious. And Gauss’s law will turn out to be particularly useful later when we combine it with other electric and magnetic field equations.
Gauss’s law is the mathematical statement of our observations in Section 24.2. There we noticed a net “flow” of electric field out of a closed surface containing charges. Gauss’s law quantifies this idea by making a specific connection between the “flow,” now measured as electric flux, and the amount of charge.
But is it useful? Although to some extent Gauss’s law is a formal statement about electric fields, not a tool for solving practical problems, there are exceptions: Gauss’s law will allow us to find the electric fields of some very important and very practical charge distributions much more easily than if we had to rely on Coulomb’s law. We’ll consider some examples in the next section.