Are there exceptions to Gauss's law?

Question

The textbook I am reading claims that Gauss's law is a fundamental law of nature, however is that really true? For example, would it hold up if inside the closed surface there was a negative charge and outside a positive charge?

Answer 1

The trick is that Gauss's law tells you about the flux of the Electric field through your surface, it doesn't actually tell you the electric field directly.

So regardless of your charge distribution and any weird closed surface you may choose, Gauss's law will correctly give you $$\oint \vec{E} \cdot d\vec{A}=\frac{Q}{\varepsilon_0}$$

But $\oint \vec{E} \cdot d\vec{A}$ is actually sort of useless to know unless we know the problem has a symmetry such that $\vec{E}\cdot d\vec{A}$ must be the same everywhere on the surface (if we choose the right surface), and we want to know what the magnitude of E is at that surface. That is how Guass's law is typically applied. But without that symmetry it just gives us the flux, and I'm not sure if there are any cases where the flux of the field through some arbitrary surface is actually useful for anything.

Answer 2

I wouldn’t use the word “fundamental” but yes, Gauss’ law appears to be a law of nature. It does hold even in the case you described, with a negative charge inside and a positive charge outside.

Any field lines from the positive charge will go in one side and out another side. The field lines to the negative charge will go in any side but terminate on the charge instead of leaving.