Gauss' Law of electrostatics is an amazing law. It is extremely useful (as far as problems framed for it are concerned :D. I do not have a real world-problem solving experience of using Gauss' Law).
However, I don't really understand why it works. What my doubt is, is this:
In the God-equation:
$$ \oint_s\vec{E}\cdot d\vec{s} = \dfrac{q_{enc}}{\epsilon_\circ} $$
$\vec{E}$ is the field at the area element $d\vec{s}$, and that field
is due to all the charges in the configuration. But the $q_{enc}$ is
the algebraic sum of only those charges that are enclosed within the
Gaussian surface. That is, to the net flux, only those charges affect that are within the surface. The contribution of the outside charges somehow cancels out. Why does it work that way?
If I knew a lot of advanced math and vector spaces and such fancy stuff, I wouldn't have asked this question. I am looking for an intuitive way of understanding its proof – if it is possible to understand without that depth of math. I do know some basic integration and surface integrals and all that (otherwise I wouldn't be able to use Gauss's Law, obviously!). So, if your answer involves these things, I'd be fine. But please don't bring in the divergence theorem etc. I tried to decipher the math behind it on wikipedia, and didn't understand a thing.
Best Answer
I can give you an intuitive view from a physicist.
Charges are the sources and sinks for the electrical field. Consider the extreme case where the volume enclosed by the surface is empty space, so no charges. Then any field line that enters the volume must exit the volume somewhere else. Thus, the integral of the field over the entire surface is 0.
If field lines only go in, but none come out, there must be a negative charge inside the volume (field lines end on negative charges), because the surface integral is negative. Vice versa, if they only come out, there must be a positive charge inside. It does not matter where the charge inside the volume sits exactly, because the same "amount of field lines" must come through the surface somewhere.
In my first semester the way they taught it to us was actually by comparing it to fish that go through a closed net inside the sea, but I don't remember what they used as analogues for the sinks and sources. Basically, if there are a lot of fish coming out of your closed net, there must be a source of fish inside somewhere.
Apologies if this is too handwavey for this part of SE.