Lots of particles in a box

Reader nanoAl had an interesting observation about the post on applying an electric charge to the earth. We were trying to find out how much electric charge would need to be applied to the earth and the moon to cancel out their gravitational attraction. The answer was suprisingly little. Here’s what nanoAl had to say:

Thats about 320 kg of electrons! I wonder how strong a box it’d take to contain the pressure from all that repulsive force, anyone know how to calculate it?

As a matter of fact, I do! Most people know that pressure is force per area. But fewer people know that it’s also the derivitive of energy with respect to volume (I’m going to play fast and loose with negative signs here, if you’re learning this for real don’t take this as anything formal):

And by analogy to the gravitational binding energy the energy required to pack a bunch of electrons uniformly into a sphere is:

Now that’s in terms of radius. We need it in terms of volume:

Hmm. I just realized I’ve changed Q to q, but that’s just a typesetting mistake that I’d rather not spend 15 minutes resetting and re-uploading. They’re the same thing – the total charge. Anyway, now we differentiate with respect to V:

Now let’s say our spherical bottle for all these electrons has a volume of one cubic meter. Plug in all our numbers from that old post, and I come up with

P = 4.58 x 1035 pascals

In English units, that’s about 66 million trillion trillion PSI. It had better be a strong bottle!

1. #1 6EQUJ5
September 11, 2008

A long time ago, Isaac Asimov came up with essentially the same thing, but his answer was a bucket of electrons.

2. #2 Uncle Al
September 11, 2008

A strong dielectric bottle. You’ve got particles in a box. Electrons are fermions. They stack energy levels, two electrons/level, and very sincerely push back upon attempted compression.

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

320 kg/m^3 electrons is 583.3 moles/cm^3. Not an optimistic number for having more slots than occupants/2. Dense packing of small atoms is 0.2925 mole of carbon atoms/cm^3 (diamond).

3. #3 Sam
September 11, 2008

While we are on the subject of electrons and questions about them, there is one that I haven’t got my head around yet. I have only a lay-understanding of quantum mechanics (only just finished A level physics and read some Hawking and Greene books), but I think I can just about cope with the wave-particle duality thing.
BUT if electrons are said to exist in ‘probability clouds’ and wave-functions, does that only concern their whereabouts in a time period?
I guess what I am trying to say is, if you could freeze an atom in time, would the electrons around it look like clouds or particles?
I don’t know if I phrased that very well, but if anyone has an answer I would very much like to know it please. It has been puzzling me trying to picture it, that’s all. Thankyou!

4. #4 DA
September 11, 2008

Sam:
Having spend a fair amount of time struggling in graduate school trying to come up with an intuitive picture for the quantum world that is easy to picture, my conclusion is you can’t. The problem, in my opinion, is language. A word like particle carries with it a very restrictive, traditional definition. So does wave. The trouble is that neither word explains what an electron is. But there are certain attributes of each that the electron possesses, so physicist use those words.

The answer to your question, as with any quantum question, depends on what you mean by “look.” How you measure the electron matters. Think of the double slit experiment. If you “look” using one way (a CCD array after the double slits) you observe a wave phenomenon. If you “look” with a detector that can distinguish between the slits you observe a particle phenomenon.

So, how are you looking at your electron? If we had the capability of “seeing” an electron in orbit (an methods are being developed to do something approximating this — look into some attosecond laser generation experiments), you would see a dot in space with a definite position. But that is only because you looked for it. Before you looked at it, it was best described as a wavefunction with a probability density of a certain shape around the atom (see a chemistry text for visualizing what the orbitals look like).

5. #5 Sam
September 11, 2008

I think you answered my question, thanks.
I was just wondering if electrons do in fact exist as tiny, tiny points of charge or if they exist more like waves of charge.
I appreciate that such things as how you look at them will alter their position, but I did mean if you were somehow shrunk small enough that an electron would be visible to you.