| MadSci Network: Physics |
Dear Steve,
Thank you for your question. If I understand correctly, you are asking what would happen if we were to start piling up electrons that are not very energetic in an enclosed area. As you correctly observe, it is quite possible to bunch electrons that are very energetic, for instance in radioactive radiation. We'll first look at this case, and then go for the case in which the electrons are not very energetic.
When we say "very energetic", we mean something specific in this case: the speed of electrons is so large that it is close to the speed of light. So, we are talking about points of negative charge moving at light speed. Now, when electrons move so fast, the force they feel is magnetic rather than electric. A more detailed explanation is in Ref.[1]. One way to think about it is (oversimplified) that the electrons move so fast that time dilates for them, moving more slowly. The repulsive force from the electric charges around them is similarly reduced. The magnetic force does not cause the electrons to repel each other. Hence, the faster electrons go, the less they repel each other and fly apart. This is crucial to particle accelerator design.
Okay, but now we come to your question: what will happen when they move slowly, and feel the full impact of the electric force? They will fly apart, and with a rather high velocity. To give you an idea we can compute the energy and forces involved in a thought experiment.
Imagine you take 2 cubes of sugar of 1 gram and put them 1 cm apart. Now, somehow, you succeed in removing all the matter except the electrons instantaneously. We can compute how many electrons we have left in either cube by computing how many electrons there are in a molecule of sucrose [2], the main constituent of sugar, and how many molecules of sucrose there are in a cube of sugar. Sucrose is C12H22O11. Counting 6 electrons per C, 1 per H and 8 per O [3], we get a grand total of 72+22+88=182 electrons per molecule. We can compute the molecular weight of the molecule in Atomic Mass Units by summing the weights of the individual atoms. Such a molecule weighs 12x12+22x1.0*16x16.0=342 AMU, or 5.68×10-22 g [4]. Hence, our gram of sugar contains 1 g/5.68×10-22 g=1.76x1021 molecules with 182 electrons each gives 3.2×1023 electrons for each lump.
The force between the electron clouds can be estimated by approximating the clouds with point charges, equal to the charge of 3.2×1023 each. This gives a charge of 51x103 C, with the C standing for Coulomb, the unit of charge [5]. We can use Coulomb's Law [6]:
F=kC(q1q2)/r2
Here, r is the distance and kC is a constant equal to 8.988×10-9N m2 C-2. This gives a force of 2.3×1023 N. This is indeed an electrostatic force, as you suggest in your question. This is about one third what the Moon would weigh if you were to put it on the surface of the Earth[7]!
Clearly, the forces involved here are massive-there is no feasible way in which you accumulate large amounts of positive or negative charge. In reality, there are charge separations, but they involve amount of charge that are quite modest-not 50.000 C as in our thought experiment, but rather a few billionth of a coulomb. Incidentally, plasmas, which you mention in your question, are examples of gases in which due to some reasons, there are significant separation of charge (though typically only a tiny imbalance between the positive and negative charge exists, perhaps as little as one in a billion or less).
Regards,
Bart Broks
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