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.
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