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Gas Container: Newsletter Challenge (06/28/05)

Posted June 28, 2005 7:00 AM

The question as it appears in the 06/28 edition of Specs & Techs from GlobalSpec:

You come home from a business trip, excited to see your family, and when you walk in the door you hear the familiar bickering you've come to love so well. Your son and daughter are at it again, this time arguing over a physics question your daughter had on a final exam. Your son says, "Dad, say you have a spherical container of gas. The molecules in the container resist crowding and get as far apart from each other as possible, right?" You carefully agree, but then he says, "So the molecules distribute themselves uniformly throughout the volume?" You say "Yup, that's right. So what's the problem?" Your daughter chimes in, "Well, Dad, say you also have a solid sphere of copper and charge it with electricity. The free electrons also try to resist crowding and get as far from each other as possible, but this is different than the case of a gas." Your son jumps in with "It's the same thing, wing nut!" "It is not, you little twerp!" your daughter spits back. "It's a different kind of system!" You briefly consider heading back to the airport to find somewhere else to go, but then settle them both down to teach them a little physics. What do you say?

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Anonymous Poster
#1

Kids talking physics?

06/28/2005 1:46 PM

Mine are usually watching Sponge Bob or playing Nintendo.

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Power-User

Join Date: Jun 2005
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#2

not quite sure

06/28/2005 3:37 PM

I'm not quite sure what the question is asking, but the net effect is the same, relatively uniform distribution of gas. However, when you charge a conductive sphere full of gas, there is no net effect on the gas inside because whether charged or non-charged relative to the conductive shell, there exists no electric field gradient to cause repel or attract forces between the gassious atoms inside. Also, the gassious atoms in both cases are not stationary but moving and colliding with the side and each other at approximately 500m/s. It is only through the law of averages that you can consider the distributution through time as essentially uniformly distributed.

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The Engineer
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#3
In reply to #2

Re:not quite sure

06/28/2005 4:04 PM

I think the question is asking why free electrons aren't evenly distributed throughout a uniform conducting sphere (I think they're all on the surface of the sphere instead). Analogies like flowing water and gas are good from helping to visualize certain problems in Electrostatics, but should be kep in the context of an analogy, not a law.

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Active Contributor

Join Date: Jun 2005
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#9
In reply to #3

Re:not quite sure

06/29/2005 4:35 PM

The two cases are not really analogous. In case of a charged metallic sphere the behaviour is governed by the fact that the electric field inside the solid mass has to be zero. This inturn is because of the fact that there are "free electrons" in metals and if the electric field is not zero, the electrons would move to make the field zero. So the excessive charge lies only on the surface. And the charge gets distributed uniformly on the surface only in case of spherical surface, charge accumulates more densely on pointy parts of a non uniform metallic surface. But that is another discussion. As far as gas molecules go, they get distributed uniformly over the volume. The molecules move around randomly, and the probability of them being distributed all over uniformly is more than the probability of molecules accumulating in one corner. I don't think it has anything to do with lowering the energy of the system (if I remember correctly from high school days!)

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Power-User

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#4
In reply to #2

Re:not quite sure

06/28/2005 4:26 PM

The question stated that it was a solid sphere of copper. It also stated that the sphere was "charged" with electricity. I am not an Electrical Engineer, but I don't think these two systems can be considered the same. The forces at play are different. The force that keeps the gas molecules uniformly distributed are a constant replusive charge associated to the electrons. The electrical charge associated with the solid copper sphere, is a flow of electrons on the surface of the sphere. The free electrons in the center of the sphere should not be affected by a current passing through the sphere or by electrons flowing around on the surface of the sphere. When the current stops flowing, aren't there the same number of electrons in the sphere as before current was applied? Otherwise would we not change the net charge of the copper? Does this have to do with the electon cloud surrounding the copper atom that has a degree of uncertainity of the exact location of an electron at a given instant? Hey I am just a Chemical Engineer reaching way back to that one electrical class in college.

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#5
In reply to #4

Re:not quite sure

06/28/2005 4:53 PM

I failed to see that it was a solid sphere. Now the charge refers to the amount of free electrons, or missing valence electrons, one of the two. Now it is true, that air pressure and voltaic 'pressure' or potential are very analogous in terms of their wave properties sound/RF and potentials/pressures in a circuit environment. The major difference between the two distributions is that the gassious molecules are apart simply because their speed is too great to allow a liquid or a solid to form, but also remaining a liquid or solid and not forming into a singularity because of the like-charged electrons of the molecules themselves. Electrons, however, experience a uniform repulsion to each other as an edict of nature. They will repel as long as there is no force keeping them together. In the case of the sphere, the free electrons will move to the surface of the sphere, spacing themselves equally apart, but not leaving the sphere because there is no conductive path past the surface off the sphere. This uniform distribution is analogous to air only by the law of averages, but like I said, air is moving around, whereas electrons are not in this example. In short though, the electron/hole density does have a net change in the system provided by the source of the charge.

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#6
In reply to #5

Re:not quite sure

06/29/2005 1:47 PM

I agree with you on the gas system, but I don't understand enough about the solid copper sphere. If it was "charged" with electricity, and electricity is actually the flow of electrons from negative to positive (or vice versa depending on the accepted theory now a days)with DC voltage, and this flow occurs on the surface of the conductor, ie the surface of the sphere, are the electrons in the center of the spere affected? I am assuming the sphere is in a circuit with two electrodes attached to the surface opposite of one another. When the sphere is "charged", does that refer to making change in the total number of electrons in the sphere, or is it only "charged" when current is flowing?

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Power-User

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#7
In reply to #6

Re:not quite sure

06/29/2005 3:16 PM

when something is charged, it is irrespect to the flow of current/electrons. As mentioned previously, voltage/charge refers to the relative density of free electons (extra valence electons relatively free to move to adjacent atoms in a conductor). For example, a battery will have a fixed voltage or charge differential between it's positive and negative terminals due to the chemical deposition of electron enriched ions to one terminal (anode), and electron deprived ions to the other (cathode). In this case, imagine that there are more free electrons in this sphere with respect to the surrounding environment. In this case, they will be repelled by the presence of each other, and attracted to the effectively neutral surroudings (ground, earth, building.. ect.) Point being that it is charged with respect to the frame of reference and in that case, the electrons will be injected into the spehere by a like force of higher charge density, thereby resulting in electons on the outside. Now the electrons on the inside of the sphere, the ones on the metal atoms are unaffected because within a sphere with charged exterior, there is no gradient, no change in the elctric field within the sphere to cause any movement, like a mesa/plateau (sp?). It's an interesting fact that spherically arranged point charges are similar to a gravitational force as well in that if you find yourself floating in the center of a hollow planet, the gravitational forces equal out no matter where you are inside, so you will not be pulled to any side, sorry bub. In like manner, the electrons in the sphere that are part of metal atoms will not move.

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Power-User

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#8
In reply to #7

Re:not quite sure

06/29/2005 4:19 PM

OK, if the electrons injected into the sphere accumulate on the surface, why would there not be gradient from the core of the sphere to the surface? And if in every system seeks equilibrium, why would the extra electrons eventually be evenly distributed from surface to core? I assume the answer to this is that there is not a gradient from core to surface to equalize? What if an anode was imbedded in the center of the sphere, if a cathode was introduced to the surface would the electrons flow through the sphere?

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Active Contributor

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#10
In reply to #8

Re:not quite sure

06/29/2005 4:50 PM

shooter you raise a good point. But in case of a metallic sphere, the whole sphere has the same electric potential, even though all the charge is accumulated only on the surface. So there is no "gradient" or difference of potential between core and surface, so no charge would flow from surface to core.

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Power-User

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#11
In reply to #8

Re:not quite sure

06/29/2005 5:00 PM

There is no gradient because of the plateau analogy I used. Electrically, the system is at a static equilibrium with a uniform charge density, but an uneven charge distribution. The key here being that density and distribution are different. Imagine that the sphere as a pressure container. In this case, the voltaic pressure (charge) is equivalent to air pressure, bringing this whole topic full circle. The container, from a macroscopic perspective, has a uniform pressure distribution, which is analogous to electric charge density. However, this voltaic pressure is achieved at equilibrium by having the electons apart from each other on the surface of the sphere, a biproduct not included in the air analogy. The core is at the same potential as the surface. In this case, the system is in steady state, but not equilibrium. In equilibrium, the relative density of electrons would be very near zero, with no charge density in respect to ground, and no electric field gradient. If an anode was embedded at the center of the sphere, and a cathode at the surface, you end up effectively with a very short and very fat cone shaped wire, a short circuit, but assuming it is not superconductive, there will still be a potential difference between the two points, causing current to flow from surface to core. Now in this case, the anode is the sphere and the cathode would be earth. If the two come in contact the electrons will move from the sphere to the earth. Another interesting point to note here is that this story did not mention the presence of an earth ground or cathode. In close proximity, the electrons will be attracted towards the earth, and the 'holes' or absence of a complete valence set of electrons will be drawn to the gap as well. The electrons will then be grouped to one side, but there is still no steady state gradient because their imbalnace will counteract the charge presence in the earth, maintaining charge density but not charge distribution, not to mention creating a poor capacitor.

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Participant

Join Date: Jul 2005
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#12

What force dominates?

07/01/2005 11:14 AM

I think that in the gas model, the brownian force dominates the position of the molecules - distributing them randomly in the sphere, like the proverbial drunken sailor. In the metal sphere, the electron's position would be dominated by the electro-static forces which make them repel eachother as far apart as possible - on the surface of the sphere, kinda like the chess club at the Jr. high school dance floor.

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Guru

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#13

Gas container

07/08/2005 5:49 AM

In ordinary gas the molecules are not charged so do not repel each other at long distance (relative to dia), but when close to each other the electron clouds of the molecules repel, causing the molecules to bounce off each other and the walls of the container. This results in even distribution. (If the molecules always repelled each other, as suggested by one of the contributors, there would be increasing gas pressure and density from the centre to the surface). Gravitational attraction is quite negligible – I estimate that for air molecules (MW ~ 30) each with 1 extra electron, the electric force exceeds the gravitational by a factor about 10 to power 33, and that's a lot. In a copper sphere some of the electrons of each atom are not tightly bound but are free to wander thru the metal as an "electron gas" moving about like molecules in an ordinary gas (thermal motion). The positively charged copper ions are fixed in the metal lattice. If the sphere is uncharged, these electrons are evenly distributed, as in any volume there is zero net charge. If the sphere is negatively charged (additional electrons), there is a net repulsion between the electrons causing them to move towards the surface. I imagine they won't all go right to the surface, but form a layer of thickness decreasing with net charge and increasing with temperature. It's also possible for the sphere to be positively charged by removing electrons. This causes the electrons to form a central ball, dia. depending on net charge and temperature. An analogy is a planet made up of a porous solid plus some gas – the gas would form a central sphere.

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#14

And the Answer is...

07/12/2005 12:47 PM

As written in the 7/05 issue of Specs & Techs from GlobalSpec:

Well, your daughter will have done well on her exam. The two systems are indeed different. You might think that electrons, like gas molecules, would spread themselves throughout the volume to give themselves as much elbow room from their neighbors as possible. In reality, however, the charge crowds near the outer surface of the sphere. Why such a significant difference in behavior? Because the gas molecules interact only with their nearest neighbors by bumping into them and jostling around until they've spread out. Their interactions are over short distances with their neighbors. Electrons, on the other hand, interact with all of the other electrons via their charge fields. A given electron can exert a force on all the other electrons, even without being near them. It maximizes its distance not just from its nearest neighbors, but from all the electrons in the copper sphere. In a sense, it accepts a few close neighbors in exchange for getting as far away as possible from all the others. This is the case when the electrons crowd near the surface of the sphere.

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