Charge Distribution on a Sphere: Exploring Gauss' Law

[Cyrus]

Hello, Question again.

If we have a conductor, then according to Gauss, all the charged particles will migrate to the surface where they are furthest away from each other. This seems to make logical sense. But what if I have a charged sphere and now every particle on the surface of the sphere is charged. I decided to add one charged particle directly in the center of the sphere. Wont it just stay there. It won't migrate, because there is equal charge all over the surface of the sphere, so it will be repelled in all directions from every piece of the sphere, and the net force will be zero, so it should sit there.

 

[kdkdkd]

first of all, it is impossible for charge to reside in the centre of a sphere. moreover, you have neglected the trillions of charges present in the sphere posessing free energy, which can affect your highly idealised case. it is also not possible to "pick and place" an abstract thing like charge in a place you want.

 

[Tide]

Cyrus,

Yes, it will sit there in the ideal case. However, the slightest thermal motion of the charged particle will set it off from the center and it will cause the charges residing on the surface to rearrange themselves in response to its electric field. The induced field will then propel the charged particle to migrate to the surface and symmetry will again prevail.

 

[Cyrus]

OH I see, we were just told that the particles would be all about the center, and so I wondered about this kind of senario. I don't think general EM physics will go into much detail on Gauss so I posted my question here. From what I understand, only one small particle at the surface can have charge. So with a sphere, they will all have equal charge at the surface. Is this like saying, all the ions inside the soild will migrate to the surface, or would it more likely be that all the atoms inside the sphere will take all the electrons they need to be neutral, and so the atoms that are at the surface find themselves in excess or a lack of electrons, so that is how the charge "migrated" to the sufrace. The actual atoms inside don't move, its just the electrons that move the charge.


Also, Is it possible for all this charge to go to one single atom at the surface? Because if there is charge, and according to Gauss only the atom at the very surface can have charge, then if I want to add more of the same charge, there is now a conflict. I can have only one atom of charge along the very surface. So would adding more charge cause the individual charges of each atom to increase.

 

[ehild]

Hello, Question again.

If we have a conductor, then according to Gauss, all the charged particles will migrate to the surface where they are furthest away from each other.

No, Gauss hasn't said that.
For example, all the particles of a metal are charged as the metal consist of positive ions and negative electrons. Do you think that all the particles of a solid metal bulk are on the surface? Or at least all the free electrons are on the surface?

This seems to make logical sense. But what if I have a charged sphere and now every particle on the surface of the sphere is charged. I decided to add one charged particle directly in the center of the sphere. Wont it just stay there. It won't migrate, because there is equal charge all over the surface of the sphere, so it will be repelled in all directions from every piece of the sphere, and the net force will be zero, so it should sit there.

If every particle on the surface is charged this does not mean necessarily that there is a net charge on the surface. There can be positive and negative charged particles and their charges might cancel. You might mean that the metal sphere has got a net charge and this charge is distributed on the surface, don't you?

So assume that you add some extra electrons to a neutral metal sphere. They will be distributed on the surface. Now you put a negatively charged particle in the middle. You are entirely right, it will not move. But the conductor is called conductor because it contains free negative charges, free electrons. These electrons around the negative particle will be repelled by its negative charge, they will move outwards as long as they can, till they reach the surface. So the charge of the particle in the middle appears on the surface while the particle stays sitting in the middle. Are you convinced or not yet?

ehild

 

[Cyrus]

Ehlid, but if it is exactly in the center, then there is nothing to cause it to move away, wouldent you agree. It is being pushed in all directions from the charge along the surface of the sphere. So the net force on that center charge is zero. And as tide said, it would remain at rest until some thermal motion caused the particle to move.

 

[ehild]

Ehlid, but if it is exactly in the center, then there is nothing to cause it to move away, wouldent you agree. It is being pushed in all directions from the charge along the surface of the sphere. So the net force on that center charge is zero. And as tide said, it would remain at rest until some thermal motion caused the particle to move.

Have you read my post at all? I never said that your charged particle will move. The charge is an attribute of the particle in the middle, not the particle itself.

So the charge of the particle in the middle appears on the surface while the particle stays sitting in the middle.

An amount of charge equal to the charge of the particle in the middle will appear on the surface.

ehild

 

[Cyrus]

Oh I am sorry Ehlid, I read your post last night at 3:34am, and I misread it. I see what you mean now. Can you explain how the charge would be represented at the surface then? Because as you say, "These electrons around the negative particle will be repelled by its negative charge, they will move outwards as long as they can, till they reach the surface. So the charge of the particle in the middle appears on the surface while the particle stays sitting in the middle. Are you convinced or not yet?
"

Im having trouble seeing how the charge will move internally to the surface. Could you explain how that would occur please? I thought that the atom in the center would be missing an electron and thus have a charge of 1e-, so that would mean that an electron from the surface migrates to the center to make the center overall neutral, but now the surface has had to give up one electron, and so the surface has a charge of 1e- at some spot where that one atom gave up its charge.

 

[ehild]

I thought that the atom in the center would be missing an electron and thus have a charge of 1e-, so that would mean that an electron from the surface migrates to the center to make the center overall neutral, but now the surface has had to give up one electron, and so the surface has a charge of 1e- at some spot where that one atom gave up its charge.

This is a good picture and it is almost the same I said, only I spoke about some extra negative charge. There is only some confusion concerning the signs. An electron has got one negative elementary charge, -e (e=1.6x10^19 C ), so an atom that misses one electron is positively charged by e.

A metal entirely consist of positively charged ions and free electrons, There are positive ions everywhere surrounded by clouds of electrons in all metals.

In equilibrium, the electric field strength is zero everywhere inside the metal. The macroscopic charge density is also zero but this zero charge density is the result of even distribution of positive and negative charges. If you place an extra charged particle of charge dQ to the center there will be a Coulomb field around it for a very short time, till the new equilibrium is established. This field of the extra charge attracts the opposite charges and repels the identical ones of the metal.
Assume that the extra charge is negative, -dQ: Its field pushes the electrons of the metal outward. The electrons can move freely in the metal, and they will move till the new equilibrium charge distribution is established: the charge density becomes zero inside the metal, and the original charge of the surface has changed by -dQ. The same happens only in the opposite direction if the extra charge is positive. There will be a Coulomb field around it, this electric field attracts the free electrons in the metal towards the center. They will move till the inner part of the metal is neutralized. If an electron moves close to the charged body in the center it will be missing at its original position, and its place will be filled by an other electron and so on till the last electrons come from the surface, making it more positive by dQ (as the lack of negative charge means the presence of positive charge). The result in both cases is that the charge of the particle in the center appears (or turns up, I do not know what is the appropriate English word) on the surface while the particle remains in the center. And this is the result of migration of the free electrons of the metal in the Coulomb field of the charged particle placed in the center.



ehild

 

[Cyrus]

Oh, I never knew that all the atoms in a metal were all positive ions. I thought that all the atoms in the metal were simply neutral.

OHhhhhhh, now I see what you are saying ehlid. My problem was not seeing it as metal ions surrounded by electrons. I was looking at the situation as just a bunch of neutral atoms.

Now I have another question for you :-). Charge is defined at the quantum level, in terms of intiger numbers of e. So if I have some sort of a shape, does this mean that the MAXIMUM charge that I could give to that shape would be determined by the following:

1. The surface area of the shape
2. The number of atoms that will cover the surface area.
3. The number of electrons the atom can hold in excess or a lack thereof.

So, would these three things determine the limit of charge I could put on an object. If I were to put a charge of 1e in the center, then a charge of 1e has to be taken away at the surface. But I cannot break down 1e between all the atoms, this is not possible. So I can only take away 1 electron from 1 atom. So at the surface, the charge is really at only one atom. The only time that the surface would be totaly charged is when I have taken away or added so many electrons, that every single atom on the surface either has the same number of electrons missing, or the same number of electrons in excess per individual atom.

Thanks so much for your help ehlid! :-)

If I had sphere with the same charge all around it. Then that means the charged particle at the direct center will be pulled equally from all the charged particle on the surface and thus will cancel out to zero! However, if we now look at that same charged particle, it will want to be attracted or repelled from the other particels directly around it, so the NET force, will just be the force of attraction or repulsion of the particles next to it, since all the charges at the surface produce no net force in any direction. So it will continue to swap charge between each new atom it encounters until it reaches its terminal point at the surface!

Furthermore, becuase the force is a 1/R^2 force, it does not have to be at the center for all the forces to cancel out with respect to the sufrace. If we choose any point inside the sphere and sum the attractive or repulsive forces due to the surface, we should find it to be zero. So if we place that charge ANY where in the sphere that already has an evenly distributed charge, the only dirving force will still just be the attraction or repulsion of its neighboors, and it will migrate to the surface!

All is well in the world again yippie ;-P

Thanks a lot for this help ehild. I never even considered thinking of a metal as positive ions. I am trying to find something in my chemistry book or online to help explain how this works. If there are all positive ions, then the only thing to keep them together would be the free electrons I guess, and this would take on some form of sharing in order for there to be bonds. What kind of bonds would this be, Ionic is between a metal and nonmetal, so this couldnet be the case, as we have nothing but metal ions. I suppose any thing that is a metal and nonmetal combination would not be a conductor, because you would no longer have a group of positive ions. So only some combination of positive metal ions can be a conductor? Edit: Ah I found something online, it is the metallic bonds that hold the atoms together. This is acomplished via the sea of electrons.

Edit: I thought about the situation as positive ions again. If the surface charge contriubtes no force to the charge at the center, then it provides no force to move the charge located at the center. Let us asume that the charge at the center and surface are both positive. So the charge at the center will only be repelled by all the charges that do not lie on the surface. But these are all positive ions as you stated, and they are distributed around it all equally, so here, they too should provide no net force to move the charge at the center. However, would the reason that the charge is able to migrate be due to the fact that the electron sea is in constant motion, so at any instant in time, there might be slight charges in anyone direction. And due to these slight charges there is a small net force that, over time, causes the charge to reach to the surface. The random motion of the sea of electrons might cause there to be a slightly higher positive charge to the left at any given instant, so the particle at the center will move to the right. Or is this incorrect.

I attached a pic of the situation. All the black +'s contribute to the positive charge at the surface, and corespond to the blue arrows. But since this charge is equal all over the surface, they should all cancel out. I think we both agree on that part. Based on the structure of a metal, all the atoms are positive ions. So all the atoms directly next to the positive charge i placed in the center will also be positive. So this too, should cancel out all the forces in all directions, since there is the same number of red plus' around the center + charge. In terms of the electrons, would they not more or less be evenly distributed around the center charge as well, thus canceling out too, or does their nonuniformity cause the migration. Wheres the driving force to cause migration?

 

[Tide]

Cyrus,

As I mentioned before, the charge on the surface does not remain uniformly distributed if the central charge is displaced slightly from it's original position. The surface charge will not contribute any field whatever beneath the surface of a sphere ONLY if it is uniformly distributed. An off-center charge inside the sphere will cause the surface charges to migrate and, hence they produce a reaction force on the off-center charge ultimately driving it to the surface.

 

[Cyrus]

oops I made a big mistake in my picture. I was thinking of the damm ion moving again. If I put in a positive ion in that position, then effectively what the electrons in those plus atoms around them will see is this attractive force to the central + charge. Therefore, they will try to go to that + source, but as they do so, they cause a + charge to be in place where it just left, and so on and so on until you reach the surface.

Oppositely, put in a negative source, now all the elctrons around it see another electron, and they get the heck outa there because there's this newly introduced repulsive force. And they go out to the surface. I have to stop thinking about the atoms being the source of the charge and moving and remember to think that the electrons result in the + or - charges.

Tide,

I have a sphere with a uniform charge on it already. Now I place a simialr charge off center. It will migrate to the surface, but it does not matter that it is off center really does it? As we both agree, i think, the net force that the surface produces is zero anywhere inside the sphere. So its the interaction of the electrons that causes the migration. I do not understand what you mean by the surface charge will no be uniformly distributed if the central charge is displaced slightly. I thought that it will remain evenly distributed. (I guess what you mean is that once migration has fully occured, there will be one point on the surface, one atom, that now has to account for this extra dq of charge, and the surface no longer remains uniform AFTER the migration, but it is uniform BEFORE the migration takes place, with a source charge not along the center?)

 

[Tide]

I have a sphere with a uniform charge on it already. Now I place a simialr charge off center. It will migrate to the surface, but it does not matter that it is off center really does it? As we both agree, i think, the net force that the surface produces is zero anywhere inside the sphere. So its the interaction of the electrons that causes the migration. I do not understand what you mean by the surface charge will no be uniformly distributed if the central charge is displaced slightly. I thought that it will remain evenly distributed. (I guess what you mean is that once migration has fully occured, there will be one point on the surface, one atom, that now has to account for this extra dq of charge, and the surface no longer remains uniform AFTER the migration, but it is uniform BEFORE the migration takes place, with a source charge not along the center?)

No, the surface charge will not remain uniform if the interior charge is offset from the center. The sea of electrons and the surface charge will continue to redistribute themselves until there is no electric field inside the conducting sphere. Ultimately, all the "action" will be at the surface (within about 1 debye length) and that is where the "test charge" will ulitmately reside. By the time the test charge becomes part of the surface charge symmetry will once again be established.

Only when the test charge is positioned precisely at the center will it remain there. In that case, it's electric field is radially symmetric and the sea of electrons will redistribute itself (symmetrically) to cancel out the electric field of the test charge. The only electric fields will, again, be near the surface.

However, physically, that charge cannot remain at the center if there is any thermal motion and once it becomes displaced the preceding scenario will prevail.

 

[ehild]

oops I made a big mistake in my picture. I was thinking of the damm ion moving again. If I put in a positive ion in that position, then effectively what the electrons in those plus atoms around them will see is this attractive force to the central + charge. Therefore, they will try to go to that + source, but as they do so, they cause a + charge to be in place where it just left, and so on and so on until you reach the surface.

Oppositely, put in a negative source, now all the elctrons around it see another electron, and they get the heck outa there because there's this newly introduced repulsive force. And they go out to the surface. I have to stop thinking about the atoms being the source of the charge and moving and remember to think that the electrons result in the + or - charges.

I think you got it at last.

Of course, this picture is not a very accurate treatment of the charge transport in a metal, but it will do for the time being.


As for the maximum charge that can be given to a piece of metal: If you charged that piece of metal so much positive that this positive charge sucked away all electrons from the surface, that is you depleted the one-atom thick surface layer all of its electrons, this layer would not be metal any more in the lack of conduction electrons. So the surface of the metal would be one layer below. By the way, what would keep these ions together? Because it is the cloud of free electrons that keeps together the ions of a metal. So the metallic surface would be one layer below.

But the electric field around the metal of so high surface charge density would be very strong, so as it would ionize the surrounding atoms of air. You would see nice sparks of electric discharge and the metal ball would loose all its charge.



ehild

PS: my name is ehild although I must admit Ehlid sounds much better.

 

[Cyrus]

No, the surface charge will not remain uniform if the interior charge is offset from the center.

But this is a 1/R^2 force. If the particle is off center anywhere inside the metal sphere, as long as it is not on the surface, won't the net force be zero. It will be closer to the surface at one point, and feel greatest forces of repulsion when closest to the surface. But at the same token, there will be less repulsive forces allong the points of the surface that are further away from the point. However, there will be more of these weak forces to balance out the fewer strong forces, and the net force will still remain zero anywhere inside the sphere.

I attached a pic. see the off center charge feels big forces closes to the surface, the dark blue arrows, but there are fewer of them. As you go around, you have more and more of the weak skinney blue arrows, however, there are more of them, and the net vector when adding all these arrows is zero. So even if this charge is placed off center inside an UNIFORM sphere with a uniform charge on its surface, we know that it is not the surface that contriubutes to the motion of the charge. The source of migration has to be the electrons that are around the metal ions just next to that blue off center point. ( the red dots.) If it were not for the red dots and their electrons, there would not be a charge migration. Is this right ehild?

Im sorry I spelled your name wrong ehild.

"Compare this result of eq.(12-26): instead of having r, the distance beteween m and the center of M, in the denominatior; we have R, the radius of the shell. This means that U in eq.(12-28) doesn't depend on r and thus has the same value everywhere insidee the shell. When m moves around inside the shell, no work is done on it, so the force on m at any point inside the shell must be zero." - Univserity Physics Text book.

This is for gravitaional forces, but we can also apply this to coulmb forces, because its is the same as the gravitational force, exctept for the sign conventions. So thereforce, even if the point charge is off center, the surface charge produces ZERO net force on the point charge ANYWHERE inside the sphere. So as I said earlier, wouldent the only source of migration have to be a result of the electron interactions between that off center charge and its immeditate neighbooring electrons. So it would not be necessary for a thermal effect to happen before the charge would migrte, the charge would migrate no matter what, even if the thermal conditions were to remain exactly the same, as that is not the source of migration.

The sea of electrons and the surface charge will continue to redistribute themselves until there is no electric field inside the conducting sphere.

So I do not see how you can conclude that the surface charge will continue to redistribute themselves until there is no electric field. It is only the elctrons that redistriubute themselves. The surface charge plays no role in this migration.

 

[krab]

... So thereforce, even if the point charge is off center, the surface charge produces ZERO net force on the point charge ANYWHERE inside the sphere...

That is not correct. The charges at the surface do not remain uniformly distributed. They will respond to the charge at the off-centre position, and their re-distribution will result in the off-centre charge experiencing a force that will drive it further off-centre.

 

[Cyrus]

Krab, could you please explain how would the off center charge would produce a change in the distribution of the charge at the surface? I don't see how the off center charge will cause this redistribution. Is what i stated about the net force correct if we limit ourselves to that instant in time the charge is placed, before the surface has time to react and redistribute to the off center charge? But at the next instant in time, the charge will redistribute, and then the forces won't be uniform, and so there is a net force.

 

[Cyrus]

Maybe I should start with a simple case of one charge with no charge at the surface and we can slowly build up to my question. That way I can be sure I am not making wrong asumptions.

I will start off with the case where we have a buch of red ions making up the metal, and a sea of electrons holding them together. Let's say I add a positive charge at the center, the blue square. Now an electron will move into this positive source from one of its immediate neighboors. But as a result, it will leave behind a positive ion, and so the charge has migrated. But as this process continues, why would the charge migrate to the surface. It would seem like it would just constantly drift all over the inside of the sphere. Meaning, a neighboor looses one of its electrons in order to go to that attractive blue square. But now the one that give it up wants to get one for itself too. Could it not demand that electron back from the very blue dot it gave it to. Then the charge would be going back and forth, or it could be randomly drifting within the sphere. If it were to drift long enough and make it to the surface, wouldent the atom just blow the one at the surface want to give its electron up to this positive source at the surface, and now the charge started its way back down into the sphere. Would the motion of the charge not be totally random in this case?

 

[krab]

Would the motion of the charge not be totally random in this case?

I think you make things too complicated. Electrons are mobile. A free charge results in an electric field. All electrons respond to this electric field: they each experience a force = e E. They're mobile, so they begin to move. They continue to move until the electric field is zero. This happens when the electron deficiency is migrated to the surface. BTW, the itinerant electrons in a metal are not localized states; one electron can be spread over the whole surface.

 

[Cyrus]

I think you make things too complicated. Electrons are mobile. A free charge results in an electric field. All electrons respond to this electric field: they each experience a force = e E. They're mobile, so they begin to move. They continue to move until the electric field is zero. This happens when the electron deficiency is migrated to the surface. BTW, the itinerant electrons in a metal are not localized states; one electron can be spread over the whole surface.

Sorry krab, I am trying not to make them complicated . Hmm, maybe I should ask my question like this. The moment we put in that test charge in the metal, it will migrate. But during the process of migration, does it do a sort of swap as it progresses to the surface?

0 0 0 + 0 0 0
0 0 + 0 0 0 0
0 + 0 0 0 0 0
+ 0 0 0 0 0 0

If it started at the center, does it migrate by "Stealing" and electron from either 0 atom to its immediate left or right. But as a consequence, causes the one that just lost an electron to want to "steal" another electron from the one to its left or right?

 

[Cyrus]

Krab I understand what your saying. But I am trying to see how it works from a molecular standpoint. When a charge is placed, you say that all the electrons will move as a result of the force produced by the charge. But what I am trying to think is that the motion of the electrons are random within the metal. And at any given instant, the distribution is not totally equal, since they are jumping from atom to atom and wizzing around all over the place. So if this test charge is present, then all the electrons are going to be attracted to it. However, since the distribution is not the same all around, one electron might get there first before all the rest. If that happens, the charge at that one spot would go to zero, but a charge would appear in the spot where that electron just left.

 

[Simonnava]

This is a good picture and it is almost the same I said, only I spoke about some extra negative charge. There is only some confusion concerning the signs. An electron has got one negative elementary charge, -e (e=1.6x10^19 C ), so an atom that misses one electron is positively charged by e.

The elementary charge is = 1.6X10^-19 C not 1.6x10^19 C

 

[ehild]

The elementary charge is = 1.6X10^-19 C not 1.6x10^19 C

You are right, I meant it with the minus sign, it was lost somewhere by accident...

ehild