I mean whatever keeps the electrons from moving further and further apart by simply expanding the (outer) shell. So yes I mean the 'container' if you like.

My plan on this, as a first cut, is to model the container in a very simple way: basically as a nearly rigid barrier, or as close as I can get to that in a numerical simulation. So basically, this would look like a spherically symmetric very high potential barrier. Of course, I realize that this is completely unrealistic. Not only is there the issue that the structure would be made of atoms, but the surface roughness in a real container is far from a sphere at the molecular level. But, I figure one has to start somewhere.

I'm open to any suggestion, either now or after I put forth some documentation. Either way, I wouldn't use this suggestion until a second pass at refining the calculations. I just feel I need to walk (or even crawl) before i try to run.

Speaking of this issue of realism versus simplicity; last night I faced another case where I decided on simplicity as a first cut. In doing any kind of a dynamic calculation, used as a way to find the stable static solutions, we run into the need to have dissipation in the system. In a conductor, this is somewhat easy since we can make the approximation of using Ohms law as a form of heat dissipation. In this case, we don't have that ability. A realistic model would need to consider electromagnetic radiation due to accelerating electrons, and dissipation processes as electrons hit the walls, and probably some other things I can't think of right now. However, this is getting much too complicated for my purpose. A simple velocity dependent friction term is all I need to use, to get the electrons to damp down to a stable state. Again, the idea is to crawl first, then maybe run later.

As always, you bring up good points.

 

How are you doing the modeling? Is it a program or is it like something where you put in all the math and let the computer figure it out like.

 

How are you doing the modeling? Is it a program or is it like something where you put in all the math and let the computer figure it out like.

I was just about to head to bed after putting a couple of hours into this. I had a deadline at work today, so I was away from this project for several days. But, I'm back on it now, and can probably do an hour of work per day over the next week to get this done as a presentable first cut.

I'm looking at two different methods to get at some information.

The first method is basically a numerical calculation on computer (so, yes, a program) which I'm doing in Matlab/Simulink. This is basically modeling the dynamic motion of the electrons and determining their final resting places. From this we can see the electron structure/pattern in both 2 dimensions (circle) and 3 dimensions (sphere). This part is going well. I've documented the equations and did a two page writeup. And I've implemented the equations in Matlab/Simulink.

The second method is writing out the potential energy formulas and trying to get at some information through them. These equations are easy to write out, but not so easy to solve, so I'm doing them numerically in Matlab for now (so, yes, again it's a program). Aside from the numerical results from the formulas, I believe I'm able to provide some good insight just by showing the structure of the potential energy formula and making a few key observations. I've already formulated an argument that seems to make sense (at least to me), but it seems premature to present that in the absence of data to back it up.

So basically, I'm making good progress and seem to be keeping to my original planned schedule. I continue to think about my latest viewpoint, and I can find no logical flaws in it, but we need some data to prove or disprove it.

 

I'm impressed Steve. I never thought this would go this far. I hope I can get a break in my schedule to have a look at your model, but it's not looking that way till some time out.

Really cool!

 

So, last night I had another epiphany. This epiphany may have squashed my previous "epiphony" (misspelling deliberate ). Perhaps there will be an another epiphany tomorrow that squashes todays "epi-phony". However, I thought I should keep you all posted on major changes in thinking as I go forward. I've held off on putting documentation out to the forum because I know that there would be mistakes that I'd have to correct later. Best to get it right and then put it out.

The major anchor I'm holding to as I go forward is that any explanation we ultimately determine to be correct should be consistent with our normal 3D Gauss' Law, as least in the continuum limit. Initially, I felt that the idea of a shell structure was inconsistant with Gauss' Law, and so I didn't accept it. However, once I visuallized a way for both ideas to be valid, I was more willing to consider that as possible. I then did some approximate calculations involving potentials and these seemed to support the idea of a shell structure as a stable solution. What I didn't mention was that I did these calculations in two dimensions to keep things simple. I've now been doing more advanced computer modeling. While the programs are general and can be applied to 1D, 2D or 3D, I've so far restricted the calculations to 2D because it is easier to plot and visualize the results. I figured that I can get the basic concepts down, and then later do the 3D modeling and prove things conclusively. (BTW, those 2D simulations did show stable shell configurations)

Well, this is all a good plan, I feel, but I believe that I made a major conceptual mistake, and this has major implications to my viewpoint. What I neglected is that the form of Coulombs Law must change when we do modeling in sub-dimensions, otherwise Guass' Law is not valid in the space. I've now made this correction and I find that those stable shell states have not shown themselves yet.

I want to be thorough, so I'm am going forward again and trying to find stable shell structures as solutions. If they are there, I'll fall back on my previous "epiphony", resurrect it as an epiphany and try to show that the shell solutions converge to the Gauss Law in the continuum limit. If the shell structures are not there, then proving the limit is much simpler.

Anyway, I'm just mentioning this as an update. I've already made too many mistakes to be confident about any results yet, but I am confident that I have the tools, or can make the tools, that can prove an answer.

By the way, I will document and show the 2D shell structure plots in the final document. They are too pretty to leave out, and they are interesting in their own way.

Now that I have this "dimensionality" concept sorted out in my own mind, I can mention the 1D case, which is very easy to visualize. In 1D, Coulombs law says that electrons put a constant repulsive force on other electrons, no matter what distance separates them (remember 3D goes as 1/r^2, 2D goes as 1/r and 1D goes as 1). Potential is now linearly related to separation distance (remember 3D goes a 1/r, 2D goes as -log(r) and 1D goes as -r).

Now think about how electrons will be distributed between two infinite planes. Gauss' Law in 1D says that a continuous charge will distribute evenly on both surfaces. If we try to think about discrete electrons in this 1D world, and if we realize that force is not dependent on separation distance, then we see that all charges must get pushed to one side or the other side, and they must go right up against the wall without a shell structure. The only exception I see is when there is an odd number of electrons. There, it seems that one electron can go anywere in the middle without feeling any force. But, in the continuum limit, this lone charge would have its charge go to zero, and it would disappear. Note that in 3D, charges in a line can not bunch up right at the walls because Coulomb's law in 3D is different than that in 1D.

This 1D example gets to the root cause of why we are having trouble with the idea that electrons may not be able to form a shell structure in 3D. If we try to think of the analogous case of charges in a circle, we feel that there must be a shell structure (and there is), but once we open up to a sphere, the rules are different. ... or, at least that's what I think at 3:43 PM today. The 3D simulations I do later should be able to correct me if I'm wrong (yet again).

 

Steve, someone else is singing your tune, but in a different key.

http://physicsforums.com/showthread.php?t=569201

go well

 

Here is a summary of some analysis I did on the question of classical stable shell solutions existing in a spherical container.

This is the best I can do with the limited free time I have. Clearly there is room for a lot more work on this subject, but I felt that there is enough here for a submission. I'll be surprised if no errors are found, and any feedback is certainly welcome.

It will be clear that I now have a few calculation tools in place, and it may not be hard to extend those tools. So feel free to suggest any test cases to simulate, or make any recommendations on model improvements.

 

That was a major piece of writing, Steve.

It will take some time to study it thoroughly.

I may have read the intro too hastily but I am worried about the description of the 1D case.

Surely there is only one solution for a line of length L with n (n>1) equal charges?
That is spaced at L/(n-1) apart?

 

Surely there is only one solution for a line of length L with n (n>1) equal charges?
That is spaced at L/(n-1) apart?

Yes, you have to dive into it in more detail.

There is the case of 1D line of charge in a 3D world. In this case the charges are spaced out as you say, but the spacing is not L/(n-1). Remember previously I described the case of N=5, and the spacing was such that the 2nd and 4th charge were located 0.23L from the end, and not 0.25L from the end.

However, this is not what I'm describing in the document. I'm describing a line of 1D charges in a hypothetical 1D world. In 1D, Coulombs law does not have a 1/r^2 dependence but is instead a constant value. In this case potential has a |r| dependence and not a 1/|r| dependence.

We see similar examples in our 3D world. For example the well known solution for a charge distributed over an infinite (or at least large) plane (parallel plate capacitors and such).

 

Yes you are right, Steve, it is much more complicated than I thought in haste.

This will require many cups of tea to find a general force formula.

 

At the present time, another simulation with 1000 electrons is being run, but it is estimated that this will take over a week to run on the available computer.​

I mentioned in the document that I posted that I was doing a 3D simulation with 1000 electrons, and that I estimated that it would take a week to simulate. I was running this on my work computer, and I need to use my computer for work, so I had to stop the simulation. I figured I would post the result even though the electrons did not reach equilibrium yet. It seems clear that the electrons are heading to the surface of the sphere and that there is no tendency to form a shell structure with 1000 electrons.

The first plot shows the radial distance of the electrons versus time. Just about all electrons have gone to the surface by 2.5 seconds, where I terminated the program. There are still a few slow-pokes, but what happens is that once most electrons are on the surface, there is a shielding effect and the field in the interior is quite small, hence it takes a long time for the remaining electrons to move to the surface.

The second and third plots are 3D plots. One is showing the full trajectories of all electrons over the full 2.5 seconds. The other plot is showing the final positions of the electrons at the point that I stopped the program.

 

I've been watching this thread avidly since its inception. Steve has published some amazing original research in this thread. It has become clear from Steve's modelling that Gauss' Law is the correct interpretation--as we add more and more electrons to the sphere, they will tend to form a single layer, with no layering or shells.

The original conjecture presented earlier was pondering whether there was ever a point during the constant addition of electrons when the electrons became so packed together on the surface of the sphere that the force repelling the electrons away from each other would overcome the tendency to form a single layer and force other electrons into the interior.

Unfortunately, if the 1000 electron scenario would have taken a week to be completed, I don't believe we have a means of testing this, since it would probably require coulomb sized quantities, and I doubt there is enough computing power in the world to model such a scenario. Unless someone has some other idea of how to test for this, I think we have to accept that Gauss' law holds for an infinite amount of charge. Oh well. I was hoping we might find something unexpected :/

 

Another thought.

Steve's method relies on moving charges and I have been pondering the comments by Professor Francis Sears on page 22 - 24 of his 'Electricity and Magnetism' about the subject.

 

Astounding work Steve. That's longer than my Masters dissertation!

I need some clarification on a few points, but as it stands right now I have a slightly different interpretation of a few of the sections. I'll have time tomorrow to post my questions.

Until then, cheers!

 

Okay here are my questions, but first a comment or two...

1) The usual presentation of Gauss' law normally assumes a completely uniform distribution of charge. Kind of like you would get with a conducting sphere, or with a even distribution of an enormous number of electrons. In the case of the conducting sphere, even a single charge added to the sphere will provide a completely uniform distribution charge over the entire sphere, anther will indeed be not field effects within the sphere. Just a potential of 1e. Now, Gauss' sates only that , within a shell of charge, there will be no net field through a surface within that shell. It does not say that fields cannot exist in that space. For example, if we have 2 electrons in a non-conducting sphere of radius r, we still get a valid result for Gauss' law for a spherical surface of radius s, s<r, but to say a test electron brought close to either of the two existing electrons will not feel a force, would be inaccurate. In fact, the only place there would be no field at all would be in a circular surface of radius r passing through the center of the sphere and to which the line between the two electrons was normal.

2) I believe using the 3D Coulomb's law is appropriate for both cases where the electrons are restricted to 1D and 2D. I am not sure of the validity of changing the nature of the electric force in those cases. So I think the results in section 6 are valid for that configuration an provide some valuable insight.

3) I'd also like to point out that the original idea was to add electrons, maybe not one at a time, but in a more or less serial fashion.


And here are my questions:

1) In section 8, what is the radius you used for the circle?

2) Same question for section 9.

3) In section 9 (and where ever else this was done) - Why do you think using 1/r in a 2D case will be the same as using 1/(r*r) in a 3D case?

Thanks in advance.

 

Another thought.

Steve's method relies on moving charges and I have been pondering the comments by Professor Francis Sears on page 22 - 24 of his 'Electricity and Magnetism' about the subject.

Hi Studiot,

Can you give us a gist of what was said? I have no access to that text.

 

Hi Studiot,

Can you give us a gist of what was said? I have no access to that text.

I was also curious to know what was said.

Bill, thanks for your response above. I need to hold off for a couple of days before I can respond to your post above. I deliberately didn't try to read it in detail because I know I'll get drawn into it at a time that I can't spare any time. I'm sorry about that, but I know you are very busy now too, so I expect you'll understand.

Right now I'm still at work and may be looking at an all night effort to finish a proposal that is due tomorrow. I think you can tell that I enjoy this discussion, so I'll be eager to dive back into this on Friday and into the weekend. Maybe studiot will respond by then too and I can consider it all at once.

 

OK he makes two relevant comments.

Firstly he compares the electric force between two charged objects, one considered moving and the gravitational force between two masses.

The difference arises from the fact that while gravitational effect appear to propagate through space with infinite velocity, electrical (or more correctly electromagnetic) effects do not.
Let A and B represent two masses The gravitational force exerted on B by A is directed towards A. Now let A move to A'. The instant A is displaced from its original position the direction of the force on B changes, and in such a way that this force is directed at every instant towards the position occupied by A at that instant.

Now let A and B represent two electric charges. If A and B are at rest the force exerted on B by A is along the line joining them and is given by Coulomb's law. If A starts to move towards A' the force on B does not alter its direction instantaneously, but a certain time elapses before the changes in A's field have propagated to the point occupied by B. The consequence of this is that if charge A is in motion, the force exerted by it on charge B does not depend upin where A is now but where it was a moment ago.

Note this is a non relativistic viewpoint.

The second point is:

One more point should be mentioned in connection with the definition of electric intensity. Suppose we wish to know the electric intensity at a point of empty space. The field, in purely electrostatic phenomena anyway, is due to the presence of tharged bodies in the vivinity of the point. But if one places an actual test charge at the point, the forces exerted by the test charge on the original charges will, in general, cause them to move from their original positions.

Sorry I can't do the diagrams tonight.

Enjoy your weekend guys.

 

I'd also like to point out that the original idea was to add electrons, maybe not one at a time, but in a more or less serial fashion.

I was thinking of this as well, and had begun to write a post on it. But as I kept thinking about it, I came to the conclusion that it didn't matter. I believe that the electrons you add will always behave like those outliers in Steve's 1000 electron experiment--that is, they will slowly make their way to the surface of the sphere. It is only once they get there that another potential force might act on them. Namely, the local effects of the neighboring electrons. So, whether it be a trickle or a torrent, the result will be the same, just with a dilation or contraction of the time domain. At least, I think

 

I got my proposal in just a while ago, and since I dont' yet feel tired from my lack of sleep, I thought I'd take a preliminary look at these questions. Some aspects I'll have to think about more and come back to later with a fresh mind.

Anyway here is my first pass at your comments.

2) I believe using the 3D Coulomb's law is appropriate for both cases where the electrons are restricted to 1D and 2D. I am not sure of the validity of changing the nature of the electric force in those cases. So I think the results in section 6 are valid for that configuration an provide some valuable insight.

I agree that 3D Coulombs law is appropriate for 1D and 2D containing of electrons in a 3D world. One simply defines these as real world problems that can be thought about, or solved, or even done by some experimental means. So, providing that you can construct a situation where real electrons (or other discrete charges) are confined by a track in the 1D case or confined to a plane in the 2D case, the notion of having shell configurations is definitely valid.

As far as changing Coulomb's law for 1D and 2D hypothetical space, there are two aspects to this that give it validiy.

The first aspect is that these hypothetical examples provide "prototypical cases" that are relevant for the 3D case of a sphere. The benefit is that we may have an easier time solving these cases analytically and gaining some insight. For example, the 1D case of charges in a line, in a true 1D space can be solved exactly without need of numerical simulations. What we find there is that there are no shell structures, except that we discover the idea that one charge can exist in a stable state in the middle, when there are an odd number of electrons. This is analogous to studiot's mention that we can have one electron in the middle of a 3D sphere that is marginally stable with balanced forces. The analogy seems to work very well and gives some insight into how charges can be forced to the surface in the 3D case. So, it is not proof of anything, but it helps form one of two nice "bookends". The other bookend is the fact that we find that a line of charges in 3D space does have a shell structure only.

The second aspect is that we can find equivalent cases of the 1D and 2D problems in our real 3D world. The case of an infinitely long cylinder is directly equivalent to the 2D problem and the potential and field funtions are identical. Similarly, the case of parallel planes is directly equivalent to the 1D problem, and again the potential and field functions are identical. The only difference with 3D is that we can no longer think about point charges. Instead we have to think about charge density and charge distribution along the extraneous dimensions. The charge then is only discrete in the dimension of interest.

The main issue that is going on is that Coulomb's law (and Gauss' Law) are relying on a very special rule about how the field decays with distance. We can think about how the electric field lines spread out with distance. In 1D the field lines can't spread out because there is no space to spread out in to. Hence, force does not decay with distance. In the 2D case, the force goes as 1/r because the circumference of a circle increases as r. And, in the 3D case the field and force must decay as 1/r^2 since the field lines are spread over a spherical surface which has area that increases as r^2. This makes a 3D volume special in a 3D world, and a 2D area is special in a 2D world and a 1D length is special in a 1D world.

The thing I'm trying to show here is the pattern. In 1D space, the charges confined to a 1D container go to the surface. In 2D space, the charges confined to a 2D container go to the surface. In 3D space, the charges confined to a 3D container go to the surface. etc.

There is another pattern too. If you confine or constrain charges to a subdimension of the space you are in, you do get shell structures. Hence, in our 3D world, shell structures occur when charges are confined in a circular area within a plane. Also shell structures occur when charges are confined to a linear span within a line in either 2D or 3D.

In a nutshell, considerations of these alternate cases allow me to expand my mind, without use of drugs! What I walk away with is an intuitive notion of how the relation between the dimension of the container and the dimension of the space are important to the character of the allowed solutons.

3) I'd also like to point out that the original idea was to add electrons, maybe not one at a time, but in a more or less serial fashion.

Yes, that's a good point. The nice thing here is that we can discuss these types of things and do appropriate modeling to try and learn more. I take this comment to mean that I should modify the models and try to simulate this type of thing, and I agree and can do this without too much trouble.

This is a particularly good point because the placement of all charges near the center at once creates a type of explosion. The charges want to blast out from the center, and perhaps this makes it harder for shells to form.

I'll say that my intuition says that we will get the same results, but the whole point of this is to find out for sure, and not to guess, so I'll try to come up with model modifications that allow this test case.

1) In section 8, what is the radius you used for the circle?

2) Same question for section 9.

As far as I can remember, I use a radius of 1 m for all calculations. These sections definitely have R=1 m.

3) In section 9 (and where ever else this was done) - Why do you think using 1/r in a 2D case will be the same as using 1/(r*r) in a 3D case?

I don't think that 1/r in the 2D case is necessarily the same as 1/r^2 in the 3D case. Rather, I view this as a "prototypical" case to look at as a first step. I have no real basis to say that results in one case prove anything about results in the other, but I do think it is helpful to look at the simpler cases to gain insight. Ultimately, the answers to the questions in our 3D world have to stand on their own two feet (or is it three feet? ).