Gauss's law and the Electric field in an object

[ovoleg]

Hey guys, before I post I want to say that I am still learning this concept and just want clarification.

Why do the charges inside a metal or other object that are interacting(repelling for instance) eventually come to rest and not continue to keep moving around constantly repelling one another. To this we assume that the electric field on the inside of the Gaussian object is zero. All of the charge moves to the outside of the objects surface.

But I just don't see how the charges just come to rest. And what is the case when there are ALOT of charges inside the metal that are interacting.

Due to superposition, they should continue to keep moving?

Again, if I sound stupid it's because I am still learning :). Thanks guys!

 

[ovoleg]

Anyone :)?

Was what I said true at least?

 

[Tide]

I think you are misinterpreting some concepts. One can say that an object is "at rest" while at the atomic level they are not at rest. In terms of the conducting material, you are most likely thinking about electrical current which may be zero but that does not imply that the charges are at rest (they are not!).

As for applying Gauss' Law to find macroscopic electric fields associated with a conductor, you are making an assumption about macroscopic fields inside the conductor. At a sufficiently small scale the electric field is not zero (e.g. very close to an individual electron). However, if you average over a sufficiently large volume the field is 0.

 

[ovoleg]

This is what I take from the book, "When excess charge is placed on a solid conductor and is at rest, it resides entirely on the surface, not in the interior of the material. Here is the proof. We know that in an electrostatic situation(with all charges at rest) the electric field E at every point in the interior of a conducting material is zero. IF E were not zero, the charges would move. Because E=0 everywhere on the surface, Gauss's law requies that the net charge inside the surface is zero. So there can be no excess charge at any point within a solid conductor; any excess charge must reside on the conductors surface"(Young & Freedman 849)

But why do all the charges come to rest? Why wouldn't they keep repelling/attracting? I guess that's really what I don't understand?

Thanks

 

[FrogPad]

I may be off base saying this, so please don't take this as the exact reason. I'm sure someone here will correct me. But this is how I thought about it while taking EM. Ok you have a conductor, so the charges are limited to moving only about the conducting object. They are not going to fly off of it for example. Now the charges "want" to move to be in a static situation. They will take very complicated paths to get into their final resting places, but they will "rest" once they get there. Imagine if you took a bunch of different magnets of various strengths and polarities and attached a piece of string to each. If you could turn off the magnets and tie them all together (so it looks like a bicycle spoke... where the spokes are the strings, and at the end of each string is the magnet, and on the other side of the string is the knot where all the strings meet), and then turn them back on, the forces would all interact with each other. The magnets would move in paths that would be very hard to predict, but we do know that there would be a limit on their paths. They can't move past their strings. So after some time the magnets would become static. Now why do objects "choose" to get into those positions. I have no idea... and and definitely interested to know why. I thought your question was interesting, and am looking forward to someone knowledgeable responding.

 

[ovoleg]

I may be off base saying this, so please don't take this as the exact reason. I'm sure someone here will correct me. But this is how I thought about it while taking EM. Ok you have a conductor, so the charges are limited to moving only about the conducting object. They are not going to fly off of it for example. Now the charges "want" to move to be in a static situation. They will take very complicated paths to get into their final resting places, but they will "rest" once they get there. Imagine if you took a bunch of different magnets of various strengths and polarities and attached a piece of string to each. If you could turn off the magnets and tie them all together (so it looks like a bicycle spoke... where the spokes are the strings, and at the end of each string is the magnet, and on the other side of the string is the not where all the strings meet), and then turn them back on, the forces would all interact with each other. The magnets would move in paths that would be very hard to predict, but we do know that there would be a limit on their paths. They can't move past their strings. So after some time the magnets would become static. Now why do objects "choose" to get into those positions. I have no idea... and and definitely interested to know why. I thought your question was interesting, and am looking forward to someone knowledgeable responding.

Thanks for the reply. I am just curious on how exactly someone made this statement which basically derives the use of Gauss's law

 

[HungryChemist]

perhaps due to friction??

 

[rtharbaugh1]

Charges within a three dimensional Gaussian sphere quickly find a way to balance each other. Any excess charge ends up on the surface of the sphere, where there is nothing "outside" to provide a counter balance.

As a model, I think of a planet made entirely of water. Each molocule of water carries a charge imbalance due to the asymmetry of the molecular structure, with two hydrogens on one side and an oxygen on the other. In the interior, each molocule is surrounded by other molocules, with which it either is or very quickly becomes balanced. But the surface molocules are not surrounded, they have molocules on one side, no molocules on the other. As a result, they can be unbalanced in terms of charge. The unbalance results in surface tension in which waves can form.

In the ocean, there are currents of water beneath the surface, but the waves all stay on the surface. Maybe it will be helpful to think of the charges as being like waves, which always occur on the surface.

Hope this is helpful.

Richard

 

[ovoleg]

Charges within a three dimensional Gaussian sphere quickly find a way to balance each other. Any excess charge ends up on the surface of the sphere, where there is nothing "outside" to provide a counter balance.

As a model, I think of a planet made entirely of water. Each molocule of water carries a charge imbalance due to the asymmetry of the molecular structure, with two hydrogens on one side and an oxygen on the other. In the interior, each molocule is surrounded by other molocules, with which it either is or very quickly becomes balanced. But the surface molocules are not surrounded, they have molocules on one side, no molocules on the other. As a result, they can be unbalanced in terms of charge. The unbalance results in surface tension in which waves can form.

In the ocean, there are currents of water beneath the surface, but the waves all stay on the surface. Maybe it will be helpful to think of the charges as being like waves, which always occur on the surface.

Hope this is helpful.

Richard

What you said about waves makes total sense but what is the true basis for someone to say that all of the charges on the inside come to rest? Is this similar to laws of thermodynamics where they don't need to be proven? And if so, who took the step to say that all charges on the inside of an object come to rest?

I am confused on how someone derived this. But what you said about waves makes sense except the fact that if you assume that charges are either protons or electrons, and due to superposition it wouldn't matter if they are at the top or the bottom?

Or I am an idiot :( lol

 

[ovoleg]

Am I asking the question correctly or did I just confuse the whole concept?

Sorry to be a nuisance

 

[rtharbaugh1]

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

 

[erickalle]

But why do all the charges come to rest? Why wouldn't they keep repelling/attracting? I guess that's really what I don't understand?

I’m trying to guess what it is you do not understand. Are you asking:

Suppose we give a sphere a –ve charge and therefore there’s a net force on each electron towards the surface. But also at the surface there’s still a net force so why don’t the electrons just keep moving away from the surface into space?
If I am wrong just ignore this.

 

[rtharbaugh1]

I’m trying to guess what it is you do not understand. Are you asking:

Suppose we give a sphere a –ve charge and therefore there’s a net force on each electron towards the surface. But also at the surface there’s still a net force so why don’t the electrons just keep moving away from the surface into space?
If I am wrong just ignore this.

Hi erickalle

I hadn't thought of this interpretation of the question. I hope Ovoleg will tell us if this is what the question is about. However it may be that you are asking the question yourself?

I think the answer is fairly straight forward. The electrons, with their negative charge, are attracted to the protons with their positive charge. There are many more protons close to the surface than there are protons just outside the surface. So the electrons stay on the surface where they can be close to the protons.

Again, there may be more electrons than protons, so the surface may have a net charge, which may be wave-like in that as some electrons move a little closer to the center of the sphere, other electrons move a little further from the center of the sphere. These movements are very small compared to the size of the sphere, so the electrons do not rush away from the surface just as waves on water do not rush into the sky. They go a little way off, then the forces of repulsion are overcome by the forces of attraction and they fall back again.

R.

 

[rtharbaugh1]

So, back to Gauss, I suppose he may have become interested in the projection of flat geometries onto curved surfaces from his survey work. The survey work seems to have been done about 1810. The magnetism work was done about 1831. Maxwell brought Gauss's law into his theory of electromagnetism about 1864.

Gauss's law is still at the front of research into gravity today. Lee Smolin, of the Perimeter institute, in his second lecture of the series "Introduction to Quantum Gravity," 2006, says " Gauss's law survives because it is just curl of a 2-form. The rest of Maxwell does not survive because it cannot be written without a metric."

Thanks,

Richard

 

[ovoleg]

Hi erickalle


I think the answer is fairly straight forward. The electrons, with their negative charge, are attracted to the protons with their positive charge. There are many more protons close to the surface than there are protons just outside the surface. So the electrons stay on the surface where they can be close to the protons.

Again, there may be more electrons than protons, so the surface may have a net charge, which may be wave-like in that as some electrons move a little closer to the center of the sphere, other electrons move a little further from the center of the sphere. These movements are very small compared to the size of the sphere, so the electrons do not rush away from the surface just as waves on water do not rush into the sky. They go a little way off, then the forces of repulsion are overcome by the forces of attraction and they fall back again.
R.

I think this is what I was looking for :). Thanks a lot for the answer and a very detailed response. I really appreciate it and aI definitely learned something!

Thanks again guys!

 

[FrogPad]

Really interesting stuff. Thanks rtharbaugh1 for doing that research. I'm guessing you were also interested in the material :)

 

[rtharbaugh1]

Hi FrogPad

Yes, it is a question I have been working on for myself. I also learned a lot from yesterday's reading. My current study is in quantum gravity, but there is much I do not understand. Gauss's law seems so simple on the page...just a few squiggles of ink. And yet there is a universe of thought under those squiggles. Thanks, all, for the discussion.

R.

 

[erickalle]

However it may be that you are asking the question yourself?

Hi Richard.

Some time ago I was struggling with that question.
Take your example of the water waves. If the water atoms would experience a net repulsive force (just like the case of surface electrons on a charged sphere) they would take of into the sky.
In case of the electrons there has to be a counter balancing force.
This counter force is provided by electrostatic induction. As soon as the electrons are pushed a little bit away from the surface they induce an opposite charge.

eric

 

[rtharbaugh1]

Hi Eric

Induction is still a mystery to me. How does a magnet pull on an iron nail? How does a wire carrying an electric current attract to another wire carrying an electric current? I know how to measure and calculate these effects, but what is really going on at microscopic scales?

The usual explanation takes one of two equally unsatisfactory forms. In particle theory, the two objects exchange photons. In field theory, the two objects cause a sort of polarization in a sort of aether. I don't see any evidence of an aether, except it has to be there to support a field. And I am not comfortable with the idea that magnets and electric wires are exchanging photons. It's just me, I guess. I know that either of these models can be used as a basis for understanding, but what is really going on, under the models, so to speak?

I suspect that there is some underlying fundamental mechanism involving space and time which will eventually be found to cause the various forces. Maybe if we think of induction in terms of the expansion and contraction of spacetime we can make some progress. Anyway, that seems to be what most researchers in quantum gravity seem to be doing.

Nice chatting. Much to study. Wish there were some sort of magic screen to deflect all the carp and just let the true game fish through, but quantum computers haven't been invented yet.

R.