Hi Ovoleg
I should like to go into your question in more detail, and intend to do so this morning. On first glance, you are asking about the effect of superposition. The first answer that comes to mind won't be very satisfactory, perhaps. And I need to know more about what you mean when you use the word superposition.
My current understanding is that superposition is a quantum phenomena, which is demonstrable with current technology, but at scales far smaller than the size of a molocule of water. I believe I have seen the word used to describe photon interactions, and also some particle interactions.
I suspect you are probably not an idiot. I have worked with idiots, and in my experience they rarely volunteer any self-doubt. Even when confronted with the most blatent and offensive evidence of their errors, they continue to cling to the idea that they have done nothing wrong. In fact I would even go so far as to speculate that it is this idea which makes them idiots.
In any case please be patient. I want to refresh my memory on some things before going into your question about how Gauss derived his idea. In fact, I doubt that I am up to the task. Gauss was certainly one of the greatest mathematicians ever challenged by a graceless humanity. If I recall correctly, his life is illustrative of the dangers inherent in being too intelligent. I am not much of a mathematician but I have read into this question before and am willing, this morning, to try to find you some better references.
Certainly not a nuisance. Thanks for giving me something interesting to think about.
More later, if this branch endures.
Richard
later: well, I have already found myself in error. Superposition applies to any wave and is not strictly a quantum phenomena. See this site, which is rather interesting if you have a high speed connection and can watch the movies.
http://www.kettering.edu/~drussell/Demos/superposition/superposition.html
Now I will have to rethink my first answer. How does superposition and the idea of interference in waves apply to the original question?
You started by asking about why the charges inside a Gaussian sphere are said to come to rest. Tide gave you a good answer to that question. Simply, they are at rest if your scale of measure is such as to give an average of many smaller scale local charges. On average, all the little non-zero charges, say, between electrons and protons, come to zero, because as one local charge rises a little bit, the neighboring charges fall. In the water wave analogy, you might think about how sea level is said to be zero when we know that there are waves going up and down all over the place. Even tides, no pun intended. If you look close up, from the viewpoint of a child on the beach, the ocean is a very dynamic phenomena. But if you look at the ocean surface from the point of view of an astronaut in orbit, it hardly moves at all. The child is very close, so views the ocean as if it were large scale. The astronaut is very far away, so views the ocean as a small scale object.
So the answer to your question in post #4, as already given by Tide, is that small scale variations average out to zero in the large scale. Then in # 6 and again in #9, you seem to want to know how Gauss derived his idea. As I said, I don't know how, but I will see what I can find out.
R.
more later:
Wikipedia has a biographical note on Gauss here:
http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
I see that his genious was more respected during his lifetime than I first remembered. He did, however, suffer from depression, which is probably what made me class him in my mind with those of genious who found themselves having to endure a less-than-appreciative culture.
Now I will try to find out how Gauss came by his idea, which we now know as Gauss's law.
R
More later:
I find this in the Wikipedia link given above for Gauss:
"In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber; it led to new knowledge in the field of magnetism (including finding a representation for the unit of magnetism in terms of mass, length and time) and the discovery of Kirchhoff's circuit laws in electricity."
It seems that Gauss, for a variety of reasons, was interested in the geometry of curved surfaces. Perhaps it was his work extending the Danish grid system of survey which led to this interest. You may be aware that projection of a two dimensional grid (as on a flat paper map) onto a three dimensional surface (the surface of the Earth) leads to complications. See Mercatur projection. These complications are not trivial. Real property rights, such as the proper location of national boundaries, are involved. Wars have been fought over such errors. See the history of Brazil for example.
You might be interested in how I came by the water analogy. It happens that gravity, which is the prime influence on waves in water, and electromagnetics, which is the influence that determines charge distribution, both obey the inverse square law, which states that the force (gravity or electromagnetics) falls off proportionally to the square of the distance. So electromagnetic waves on the surface of a conductor and water waves on the surface of the ocean behave in analogous ways.
Now the inverse square law is pure geometry, and you may find it easy to convince yourself, as I did, that you can understand its implications. As I said I am not much of a mathematician. However, it is fairly easy, I think, to see that the area of a surface defined by radial lines from the center of a sphere increases as the square of the radius. Maybe I can find a link for that. I seem to recall seeing some nice diagrams somewhere. Try this link:
http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html
R
