What? [tex]W = \frac{\epsilon_{0}}{2} \int E^2 d \tau = \frac{\epsilon_{0}}{2 \left( 4 \pi \epsilon_{0} \right)^{2}} \int \frac{q^{2}}{r^{4}} r^{2} sin \theta dr d \theta d \phi = \frac{q^{2}}{8 \pi \epsilon_{0}} \int_{0}^{\infty} \frac{1}{r^{2}} dr = \infty [/tex] Griffith explains this infinite energy coming from the charge being required to assemble itself. What does this mean? Surely tearing apart an electron would not solve all of humanities energy problems. Or is the problem that an actual unit of charge is discrete, whereas the equation treats it as a continuous charge distribution? I'm not sure I understand.
Point charges are "exact" as long as you don't get too close. Quantum theory takes over when r is small enough.
Don't worry, you are not misunderstanding or missing anything. The fact is, we have no idea what electrons (or any other fundamental particle) "are" or how they came to be--Higgs bosons aside, those types of questions belong in the realm of philosophers. Physicists can only measure the properties of a particle or object and its interactions, develop mathematical theories to fit the data, and accept that that is the way the universe chose them to be. Noone has ever made the claim that our system of language + mathematics provides a complete description of the universe--at least not without getting laughed at ;). Randy
Well, if you drop the point-like hypothesis, you drop the problem. It is clear that the point-like hypothesis cannot be true, if only for the problem you are just reporting. There many known reasons to stop with the point-like hypothesis. The 1/r² law has been discovered at least 200 year ago on macroscopic objects. Applying it blindly down to the point-like size is meaningless. Quantum mechanics is often cited as a cut-off for the point-like hypothesis. However, the point-like problem remains in quantum mechanics just as in classical physics. The Heisenberg uncertainties just tells us that the "point" cannot be located precisely, but he point is still within the assumptions. My understanding is that quantum electrodynamics tackles the problem more closely because of the coupling between fields and particles being treated in a fully quantum way. Note also that general relativity would also come with some limit to the point-like assumption, since the electron could not be smaller than its Schwarzschild radius. I think that our curent understanding of the universe allows some people (but not myself) to talk about this topic without being laughed at.
My apologies for the late reply, but thank you for all of your responses. I take it that this is a strange quirk in electromagnetic theory. However, the mathematical argument seems unassailable, and consequently so do the results. So is the theory wrong, the math wrong, somewhere in between? I'm still a little lost as to why this happens. Thanks.
You may find Feynman's Nobel lecture interesting on this point. About 2/3 of the way down he talks about how he calculated the self-energy of the electron and determined that it was finite. AM
I haven't found Feynman's explanation very satisfying. If the two formulations are truly equivalent then there should be a way to achieve finite self energy with the field method. If not, then they are not truly equivalent, and so they are not describing the same physics. That is just my take.