charges inside a vacuum

wes

Joined Aug 24, 2007
242
I was just thinking about vacuums and crt tv's and a question came up. How would one go about figuring out how much charge can be fit into a vacuum of a given volume? Like if the electron gun on the tv just kept firing away and the electrons had no where to go, how much charge (electrons) could fit inside the vacuum? Like how many coulombs of charge? I know space charge has to be considered but I don't know how to calculate something like this.

It's not the same as figuring out the charge of a capacitor because they store charge on plates whereas this is inside the vacuum which is also measured in volume instead of area like the area of a plate on a capacitor.

jimkeith

Joined Oct 26, 2011
540
I think that the CRT does not hold an internal charge as you are suggesting--after the beam is accelerated by the HV, it strikes the screen phosphors--from there the current is conducted to common. Now it would be possible to construct such a vacuum tube for storing a charge, but the basic relationship Q=CV is not violated. At a constant beam current, the voltage at the plate (capacitor) would continue to increase to the point of field discharge.

This principle is already used in electrometers that are applied in particle physics.

BillO

Joined Nov 24, 2008
999
Hmm if it was full of electrons, it would not be much of a vacuum, would it?

But, in the spirit of the question, the tube would fill with electrons until the potential within balanced the potential the gun was capable of generating.

Or are you really asking what the volume of an electron is? In that case, again, you would probably have to define it in terms of potential as I am not sure there is a good understanding of the actual dimensions of the 'hard' component of an electron, or if it could even be looked at that way.

Joined Dec 26, 2010
2,148
If you could actually"fill" the space inside the tube with electrons the charge would be considerable, but in fact the relatively low capacitance of such a system would require an unimaginable voltage to do this.

If the effective capacitance was one nano-farad, which might be true for a large tube with a grounded outer conductive coating, then from Q=CV, to accumulate one coulomb of charge in the tube would require a potential of one thousand million volts. The glass would never take the electrical field strength , of course. With no grounded outer coating the glass would be less stressed, but the capacitance would be radically lower, so even more voltage would be required.

Since five coulombs of charge can be stored within a 1F 5V super-capacitor of centimetre dimensions, most of which is other matter than electrons, I think we can say that the tube can never turn into a full bottle of electrons.

wes

Joined Aug 24, 2007
242
too BillO, nope I was just talking about filling the vacuum with as many electrons as possible.

I do have another question though, how do figure out the capacitance of such a system? How do you figure out the Voltage potential of such a system referenced to say a ground (a neutral source ). I guess basically how did you get those number's Adjuster? Were they just numbers you kinda of dealt with before or just guesses based on experience?

I know the charge would pretty big too, something like 1 cubic centimeter would hold like 13,584 coulombs of charge if you could pack the electrons in there with a density equal to that of copper. I basically figured out how many atoms are in 1 Cubic Centimeter of copper and then figured just replace the atoms with electrons and then times that by the electron charge. I know that is huge too, I read on hyper physics that just two point charge's one meter apart would produce a force of one million tons! Even if you get 13,584 coulombs of charge inside 1 Cubic Centimeter, wouldn't it just like blow it's self apart, lol.

I did read somewhere that space charge limits the amount of charge a hot cathode can emit before it's unable to emit more, but it never said anything about exactly how much.

Joined Dec 26, 2010
2,148
The capacitance of a CRT bulb with inner and outer conductive coatings could be of the order of 1nF for a big one.
Effectively it is a glass dielectric capacitor like the Leiden jars of old: in some displays it helps filter the final accelerating voltage.

http://lowendmac.com/tech/crt_danger.html

Estimating the capacitance for a tube with no grounded outer is more approximate, but the capacitance of an isolated sphere of a similar size should give a rough guide. I assume that with the bulb "full" of electrons, we can take the inside to be conducting, and that the thin skin of dielectric glass is negligible compared to the surrounding air or vacuum.

If the capacitance of an isolated sphere is 4πεa, with the permittivity of free space taken as 8.854*10−12 F/m,
and the radius a is assumed to be 0.3m, this comes to about 33pF.

http://en.wikipedia.org/wiki/Capacitance

steveb

Joined Jul 3, 2008
2,436
I was just thinking about vacuums and crt tv's and a question came up. How would one go about figuring out how much charge can be fit into a vacuum of a given volume?
This is a very interesting question and their are many levels one could study and answer from.

There are two key things about electrons confined to a volume. First, they all have the same charge and will repel each other, and second they are fermions, which means that no two of them can exist in the same quantum state, and each must have it's own set of quantum numbers to describe it (this is the Pauli exclusion principle).

http://en.wikipedia.org/wiki/Pauli_exclusion_principle

If we look at this from the point of view of simple quantum mechanics and don't try to invoke any quantum field theory, then I would classify this question as the famous problem of a particle in a box. Why is the "box" needed? Because electrons repel, as we know from classical theory, and without a box, the electrons will not even get close enough to require quantum mechanics to help answer the question of what happens if they are brought close together.

http://en.wikipedia.org/wiki/Particle_in_a_box

However, we must now allow more than one particle in the box, which brings in the issue of using Fermi-Dirac statistics.

http://en.wikipedia.org/wiki/Fermi_energy

The particle in a box problem usually assumes infinite potential barriers to contain the particle. In such a case there is no limit to the number of electrons because the electrons will just keep filling in the allowed energy levels which extend to infinity. Each energy level allows two electrons, since there are two allowed spin states.

In reality, you can't have an infinite potential barrier, so the allowed number of electrons will be limited by the size of the box and the size of the potential barrier. A finite potential barrier means that electrons in the higher energy states can easily jump over the barrier if given any external excitation. Also, quantum theory allows the particles to tunnel through the barrier, with a finite nonzero (although often small) probability. Still, these facts don't change the maximum number of allowed electrons in a given volume, with a given potential barrier.

This question could get much more involved if we invoke quantum field theory and start squeezing electrons into a region of high temperature. But, let's leave that aside because the simpler assumptions are probably suitable to answer the question in the practical domains of interest.

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wes

Joined Aug 24, 2007
242
I think I might have found one way to figure out at least the energy needed to accumulate the charge at the density of copper.

If the potential energy between two electrons spaced 456 pm (double the diameter of a copper atom) apart is 3.157 electron volts or 5.05827e-19 joules, then that obviously means you needed to expend at least that much energy to get them that close right? Nevermind, lol, I figured you could then just scale that up to the number of atoms, lol. That is so not the right way, I forgot that each additional electron interacts with all the other electrons, however I am still not sure how you would go about figuring it out this way, let alone if it is even possible too.

Part of the reason trying to figure it out this way is I don't even know how to start figuring it out using quantum physics or do some the harder math behind it, lol, my brain would probably explode trying too. I can visualize without a problem even the weird quantum stuff and understand (maybe understand is an exaggeration, lol) the physics behind most of it except the really weird quantum things like entanglement, lol

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steveb

Joined Jul 3, 2008
2,436
If you want to think classically about this, you can up to a point. When electrons get to a density where the spacing might be of the order of atoms, then quantum effects begin to take over and the description I gave above is the viewpoint one must take. However, before that point, Maxwell's equations are the correct theory to apply (along with Newton's force law).

If you want to simplify the analysis, then think about a hypothetical spherical cavity with walls (capable of providing a large potential barrier) to hold the electrons in. If you think about it, this is not too different than having a spherically shaped conductor, or even a spherical conducting shell. In this case, the capacitance of a single conductor spherical capacitor (basically a Van de Graaff generator) can tell you a lot. Why is this? Well, the electrons will repel and try to get as far apart as possible. Hence, they will form a spherical shell, infinitely thin. This is exactly what happens with a conductor. Whether the conductor is a spherical shell of finite thickness or whether it is aa completely solid metal sphere, the electrons will go to the outer surface and make an infinitely thin shell of charge. As you add more and more charge, this will continue to happen until quantum effects begin to become significant. With a real conductor, the electrons would be ejected from the surface of the conductor long before electrons interact. However, in your thought experiment, the hypothetical wall of the container is considered to be a very high potential barrier that will contain the electrons.

So, it seems that the idea of a volume is not too critical when you think classically. It's better to think about how much charge you can put on the surface of the container. Once quantum effects are in play, then the idea of "where the electrons are" breaks down and we deal with the wavefunction that tells us information about probability distributions. Every energy level inside a quantum box has a different wavefunction. However, at least here we can think about electrons inside a box, or sphere, or any other volume, rather than on a surface of a container.

This is a somewhat simplistic view, and considers steady state fixed distribution of charge. But you have to start thinking somewhere. One can also think about dynamic situations, with particle beams and thermodynamics, and even quantum field theory as temperature increases beyond usual bounds.

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wes

Joined Aug 24, 2007
242
Well I did go back and read the links Adjuster gave and after rereading the capacitance link which I have read before but this time I saw what you were talking about with the equation you gave, along with the explanation from steveb (thanks, it painted a somewhat clearer picture I could understand, lol).

So from what I have taken in so far, The electrons that would be injected into the sphere from say a hot cathode would basically repel one another because they are of the same polarity. This repelling would cause them to move outward toward the surface of the inner sphere and then once the surface is covered with electrons trying to get out, they would also be repelling electrons back into the sphere. So over some amount of time the electrons would fill up the entire volume of the sphere on the inside. The Capacitance equation that was given represents the self capacitance of a sphere, so in the example given, a 0.3 m radius sphere would have a self capacitance of 33pf which according to wiki is the amount of charge that would be needed to raise the voltage by one volt or one unit if not using volts. Since the sphere is a vacuum, I would assume the sphere it self would be made of glass or whatever CRT's are made of (some mixture of glass and lead I think). Since Glass has a dielectric strength about 4 MV, that would be the point at which the sphere is completely charged. I guess it would actually stop a little before the breakdown point for safety reasons, lol.

So the only question I can think of as long as all this is right is how would you figure out the voltage? Once you can figure out what the voltage would be between the sphere and say earth with just 33 pf of charge, you can then figure out the total stored energy of the sphere charge by figuring out how many times that voltage can go into 4 MV (glass dielectric strength) which then gives the total charge.
33 pf x number of times Voltage into Glass Dielectric Strength.

Does any of this sound right, lol.

studiot

Joined Nov 9, 2007
4,998
This is a very interesting question and their are many levels one could study and answer from.
Indeed so!

But why limit things to electrons?

This is the very question nuclear physicists would love to be able to solve to build a fusion powered reactor.

The more charge you squash together into a smaller and smaller volume the more energy it takes.
Eventually your energy density is great enough to start a fusion 'fire'

Attempts have been made to contain these energy levels in magnetic 'bottles' by such devices as 'zeta' and 'tokomak'.

go well

steveb

Joined Jul 3, 2008
2,436
Does any of this sound right, lol.
Much of it does sound right, but one thing is a clear misconception. The idea of the electrons eventually filling the volume does not hold up in a classical viewpoint. The charges MUST go to the outer surface according to Maxwell's theory. The simplest way to see this is to realize that a sphere with equally distributed charge will have no field inside and hence there is nothing to repel the electrons to the inside. The electrons go to the outer surface in their effort to get away from each other. This actually applies to any shape vessel, but it is easiest to visualize with a sphere.

As far as your question about voltage, voltage for a single conductor capacitor is referenced to infinity, and the capacitance (spherical capacitor formula can be used) can be estimated; hence, you can calculate voltage and energy stored for a given charge.

BillO

Joined Nov 24, 2008
999
too BillO, nope I was just talking about filling the vacuum with as many electrons as possible.
Same thing to a large extent.

BillO

Joined Nov 24, 2008
999
Much of it does sound right, but one thing is a clear misconception. The idea of the electrons eventually filling the volume does not hold up in a classical viewpoint. The charges MUST go to the outer surface according to Maxwell's theory.
I'm not quite sure about this Steve. Are you saying that you can only have a hollow sphere of electrons?

steveb

Joined Jul 3, 2008
2,436
I'm not quite sure about this Steve. Are you saying that you can only have a hollow sphere of electrons?
"Can only" is a very strong qualifier, but under reasonable assumptions, yes. And, assuming the container is a sphere, of course.

I already mentioned some exceptions above, and there are others, such as placing unmovable fixed positive charge in the interior volume. But, given the assumptions of classical electrostatics, Gauss' law requires the electrons to go to the outer surface, provided the charges are free to move in the interior and provided there are no fixed positive charges in the interior.

Perhaps you have something in mind that I haven't considered, so feel free to elaborate if I'm missing a key point.

BillO

Joined Nov 24, 2008
999
Well hypothetically speaking, with an ideal sphere of finite extent with a wall of infinite potential. Lets say that we force electrons into the sphere until such a point that the Pauli exclusion principle demands that no more may occupy that single layer around the inner wall. Does this mean that we cannot get another electron into the sphere?

I'm just not sure that Maxwell's theory actually states that you cannot have the entire volume of the sphere 'full' of electrons.

Of course if there is more than one electron in the sphere, they will tend to place themselves as far away from each other as possible, and hence line the inside surface of the sphere, but once that surface is 'full' I see no problem in continuing to place electrons inside the void surrounded by that initial layer. I also see it initially as a potential energy problem rather than a QM problem.

In any case, I'd be willing to bet, if the math could actually be done, that you'd find that a second layer of electrons, and a third, and so on would begin to form due to the repulsive forces long before the exclusion principle began to dominate the nature of the electron distribution.

As you continue to add electrons, the potential energy associated with any particular electron should go up until such a point that it will increase beyond the potential that would have initially excluded that electron from the higher energy outer layers, so indeed the population of any given layer will increase as more and more electrons are added.

Again, I'm not capable of the math, but intuitively I would expect the distribution of electrons in the sphere would be more or less uniform long before there would be any fear of running out of QM states.

But... Maybe it's me that is missing something.

studiot

Joined Nov 9, 2007
4,998
Good evening Bill and Steve.

When discussing electron packing, don't forget most of what you have said is about steady state solutions, ie where the electron(s) might end up.

If you fired an electron or stream of electrons into the globe at (very) high velocity things might be different. It could take a near infinite time to reach the even surface distribution, longer than the next high speed electron might take to enter the fray.

go well and happy new year

steveb

Joined Jul 3, 2008
2,436
studiot said:
When discussing electron packing, don't forget most of what you have said is about steady state solutions, ie where the electron(s) might end up.

If you fired an electron or stream of electrons into the globe at (very) high velocity things might be different. It could take a near infinite time to reach the even surface distribution, longer than the next high speed electron might take to enter the fray.
studiot,

Yes, you are absolutely correct. That's why I was careful to mention that dynamic cases would be exceptions, and that much of what I'm saying applies to the static (or quasi-static) case.

Well hypothetically speaking, with an ideal sphere of finite extent with a wall of infinite potential. Lets say that we force electrons into the sphere until such a point that the Pauli exclusion principle demands that no more may occupy that single layer around the inner wall. Does this mean that we cannot get another electron into the sphere?
Now you are invoking quantum theory which is outside the assumption of classical theory. I discussed this above and mentioned that it is meaningful to talk about electrons inside, in the quantum case.

First of all, with infinite potential, there are an infinite number of allowed energy levels. So you can put as many electrons in as you want. It is only when we have finite potential that we run out of room.

We actually have a perfect physical example, if we talk about finite potentials. Think about a large atom, like gold for example. Here we have a finite potential created by the positive charges. There are only so many electrons that will fit in this potential. In this example, we know the electron wavefunctions allow the electron to be inside, away from the surface. Of course, the idea of a surface breaks down, but the analogy is still good.

So, with quantum mechanics, the number of electrons is limited and the idea of electrons only at the surface is the wrong image.

I'm just not sure that Maxwell's theory actually states that you cannot have the entire volume of the sphere 'full' of electrons.
Actually it does say exactly that, at least in the static case. Particularly, it is Gauss' Law that tells us this. The electric field due to spherical shells is zero for any shell of greater radius where an electron is, and the electric field is proportional to the amount of charge inside the radius of where the electron is. This means that the electric field is always driving the electron outward. Only by having all electrons at the outer radius, can we have static equilibrium. This situation actually arises in the Van de Graaff generator and the charge really is isolated to the last molecular layer or two in the metal.

Of course if there is more than one electron in the sphere, they will tend to place themselves as far away from each other as possible, and hence line the inside surface of the sphere, but once that surface is 'full' I see no problem in continuing to place electrons inside the void surrounded by that initial layer. I also see it initially as a potential energy problem rather than a QM problem.
How do you define when the surface is full? Think about how much charge needs to be there before the electrons are even getting close to each other. This could never happen in a practical case, but hypothetically, at that point I would expect quantum interactions to come in to play.

In any case, I'd be willing to bet, if the math could actually be done, that you'd find that a second layer of electrons, and a third, and so on would begin to form due to the repulsive forces long before the exclusion principle began to dominate the nature of the electron distribution.
Why can't the math be done? Thanks to Gauss' law, the calculation is trivial. Think about it. How can electrons on the outer layer repel electrons in the interior? The net effect of the charges in the entire outer spherical shells exactly cancel.

As you continue to add electrons, the potential energy associated with any particular electron should go up until such a point that it will increase beyond the potential that would have initially excluded that electron from the higher energy outer layers, so indeed the population of any given layer will increase as more and more electrons are added.

Again, I'm not capable of the math, but intuitively I would expect the distribution of electrons in the sphere would be more or less uniform long before there would be any fear of running out of QM states.

But... Maybe it's me that is missing something.
Yes, you are missing Gauss' law. This law does play tricks with our intuition. Even Newton had to struggle with these concepts when he dealt with gravity. But, now we have Gauss's law, so we can all be instantly smarter than Newton.

wes

Joined Aug 24, 2007
242
Studiot, isn't that basically where space charge arises, the cloud of charge that develops around a charge emitting surface?

Everyone elese:

I pretty much agree with what Billo said in that the charges (electrons) would fill up the entire volume of the sphere as long as the Sphere material had a high enough Dielectric strength to contain it. I assume the reason for thinking the thin layer of electrons around the inner sphere with none within the center is that since they are all the same charge, the electric field from each basically cancels out the other creating a point of zero electric field within. So from this it would make sense why it would not fill up, there is no electric field to push the electrons back further and further as more charge develops. However I think the reason it would fill up is that even though further away the electric field is near zero or zero, the electric field closer to one side would still be there somewhat. A electron would still be repelled away from another electron. So if electrons were forced with whatever energy needed inside, it would eventually be repelled back to the center or off at some angle as it interacted with not only the first layer but other electrons injected and as more and more electrons entered they would do the same thing, slowly losing energy and slowing down until settling into a somewhat localized area which would be the second layer and third and so on.

However if that is not true then what actually happens and why?

I do have a question though if it is true that it would it fill the volume of the sphere, Does the self capacitance equation still hold since we are not storing the charge on the surface of the sphere (well except the inside) but within the entire volume of it? Obviously if it just store on the surface then it does.

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steveb

Joined Jul 3, 2008
2,436
I pretty much agree with what Billo said in that the charges (electrons) would fill up the entire volume of the sphere ...

However if that is not true then what actually happens and why?

I do have a question though if it is true that it would it fill the volume of the sphere, Does the self capacitance equation still hold since we are not storing the charge on the surface of the sphere (well except the inside) but within the entire volume of it? Obviously if it just store on the surface then it does.
Well, you asked a very astute question there, and this reveals the crux of the whole matter. It is well known that a solid conducting sphere and a conducting spherical shell have the same capacitance. The reason for this is exactly what I'm trying to tell you. The charge equilibrium puts the charges at the outer surface either way.