If you'll allow me to disregard the effects of Strong Interaction, how would I calculate the Electromagnetic Force between two oppositely charged point particles that are in contact with one another? Let's assume two particles with opposite Elementary Charges: 1.60218E-19 and -1.60218E-19 and start with an assumption of a distance of 1fm... (Please forgive my pathetic excuse for scientific notation) F = q1*q2 / (4πεo * r²) That gives us: 1.60218E * -1.60218E / (1.11265E-10*1E-15^2) = -231 This means, if I understand it correctly, separating the particles from this distance would require 231 Newtons of force. Right? First: As distance approaches zero, force approaches infinity, but it can never be zero in this equation. If I'm not mistaken, the distance should be measured from center of mass to center of mass, so it never could be truly zero for massive particles. Given that, my assumption is that the distance for two identical spheres should be the radius. I get that, but where it loses me is when people say that point particles have no shape, because they exist as a point - therefore have no "size". I have also read an estimation of the radius of an electron to be about .0689fm. Do electrons have a physical size, and that's why the distance could never be zero? Second: If we're talking about two particles touching, shouldn't the Vacuum Permittivity (Dielectric Constant - or whatever term you want to use) be discarded? If so, how would we do that? What would the resulting equation look like? If not, why not? Thanks
Two point charges in contact would classically require an infinite amount of energy to separate (and in the process of coming into contact and sticking, they'd radiate off an infinite amount of energy). In quantum mechanics, particles can only exist in certain states, and you can never have a state in which two charged point particles are completely touching. When dealing with things like molecular physics where you want to calculate the interactions between polarized molecules, there are certain assumptions made about how those charges stay bound to their molecules so that there's a net attractive force when the molecules are far apart, and a net repulsive force when they get too close (these assumptions in turn can be derived from quantum mechanics, if desired, but chemists and physicists were aware of these molecular forces even in the 19th century). Coulomb's law is a classical physics law, and the only time a realistic classical physics problem talks about charges coming infinitesimally close together, those infinitesimally close charges are in turn infinitesimally small pieces from a smeared out cloud/fluid of charge, with finite forces between them as a result. Point charges are only really used as a starting point to understand the classical laws of electromagnetism, as a way of setting up calculus problems involving continuous smears of charge over a finite area, and as long distance approximations (any collection and arrangement of charges will look like a single point charge at very large distances). As for the radius of the electron, in quantum mechanics the electron may have a probability of being detected anywhere inside a given region, but at every one of those points where its probability wave travels, it also interacts and behaves like a point particle, so QM certainly doesn't remove the notion of point particles and indeed considers them fundamental. So in general it doesn't really make sense to talk about the electron's radius, but you can talk about things like the electron's average orbital radius around an atom and define that as the radius of the atom itself. There wouldn't be any reason to replace the vacuum permittivity constant, because in classical physics it's the only constant that ever comes with Coulomb's law. You may be thinking of cases where the permittivity changes, such as in a dielectric medium, but what really happens there is you apply Coulomb's law as usual with the usual vacuum permittivity, the charges in the dielectric separate slightly so as to leave the medium polarized, and you can put it all together in such a way that the net result is the same as if we treated the dielectric as a vacuum with an altered permeability; it's a special case simplification of the usual electromagnetic equations, not a fundamental property of the universe or any particular material. So really, asking why we wouldn't use a different permittivity constant for Coulomb's law at short ranges would be like asking why we even bother using Coulomb's law in the first place, the classical law was meant to apply to all possible situations involving static charge. Now as it turns out, at close ranges (especially at the femtometer range), the classical laws of physics break down and simply fail to provide an accurate description of what's happening at the subatomic level. Coulomb's law and the other classical laws of electromagnetism are really large-scale simplified averages pertaining to more complicated (but still calculable) quantum behaviour. One approach is to calculate what happens at the quantum level and then apply it to add corrective terms to Coulomb's law so that it can still be accurately applied at small ranges. This also explains how a point charge of finite energy could be formed in quantum mechanics, because Coulomb's law doesn't really apply at those ranges, and the amount of energy needed to assemble a finite point charge (in the form of a fundamental particle) is itself finite.
The distance can't be zero in practice because of the wave nature of matter. An electron is not a point particle. That's a myth that had grown out of mathematical convenience. One which ignores hard scientific evidence such as electron diffraction, magnetic dipole moment, and the Einstein-de Haas effect. It's all popscience nonsense for kids I'm afraid. The electron doesn't have a radius. It isn't a point particle, and nor is it some billiard ball. Instead it's a "spinor". And in physics, it's quantum field theory, not quantum point-particle theory. The electron is said to be an excitation of the electron field. The electron's field is what it is. And that field extends outwards. It has no limit. But there is an analogy that provides some insight: the hurricane or cyclone. Think of the electron as a cyclone, and think of the positron as an anticyclone. Maxwell referred to vortices, which you can find aplenty in gravitomagnetism. And of course counter-rotating vortices attract whilst co-rotating vortices repel. But note this: the radius of the eye of the storm is not the radius of the storm. They don't have a physical size but they do have a Compton wavelength. You must have heard of the reduced Planck's constant ħ? Think of the Compton wavelength as twice the diameter of the eye of the storm. Why not? Because particles are field. Talking about two particles touching is like saying the cyclone only touches the anticyclone when the two eyes meet.
An electron is a Dirac field, not a circling photon. But its wavelike properties make a difference, as CptBork had noted. If one tries to find the classical self-energy of a charged particle with radius r, one finds that it behaves as 1/r. For a dipole magnetic field, it behaves as 1/r[sup]3[/sup]. For an electron, both energies are about equal for r ~ electron Compton wavelength, and likewise for other particles whose magnetic moments are not much different from their equivalent of a Bohr magneton: (magnetic moment) ~ (charge)*(hbar)/(mass) When one gets to the Compton wavelength, one finds that one needs quantum mechanics to proceed further, and Schwinger, Tomonaga, and Feynman worked out how to proceed from there. What they worked out was quantum electrodynamics, the quantum field theory of electromagnetism. The self-energy behaves roughly as log(r) inside the Compton radius, and though it still increases, it blows up only at teeny teeny tiny distances. In fact, one gets to electroweak unification long before one gets there. One can do analogous self-energy calculations for electroweak and QCD interactions, and one finds that it's well-behaved down to Grand Unified Theory size scales. It's not much further to get to the Planck length, the length scale of quantum gravity, and that limits our ability to extrapolate further.
