Saketh said:
At any rate, trying to find a physical equivalent to a point charge is a futile search. Unless we're talking string theory, but I don't know much about string theory.
That's not true--and it certainly doesn't make the point charge useless. In fact, the point charge posed one of the most difficult problems that plagued classical electromagnetic theory (and later even QED part I/II, often referred to as an era with "the plague of the infinities"). Let's start out with a point charge q in a vacuum and write down an expression for the energy density of the field (in SI units) as
u(r) = \frac{1}{2}\epsilon_{0}E^{2} = \frac{1}{2}\epsilon_{0}\frac{q^2}{16\pi^{2}\epsilon_{0}^2} = \left(\frac{q^2}{32\pi^2\epsilon_{0}}\right)\frac{1}{r^4}
(Note the dependence on r^{-4})
For now, let me denote by g, the constant factor not involving r so that,
u(r) = \frac{g}{r^4}
How would you compute the energy due to the field in all space? Clearly, you would integrate u(r) over all space to get the contribution to the electrostatic energy from the point charge, as
U = \int_{all space}u(r)d^{3}r
If you are familiar with spherical coordinates, the volume element is d^{3}r = r^{2}\sin\theta dr d\theta d\phi. You can write it as d^3{r} = 4\pi r^2 dr directly (or after evaluating the azimutal and polar angle integrals...this a matter of taste). When I substitute this into the integral, I get
U = \int_{0}^{\infty}\frac{g}{r^4}4\pi r^{2} dr = \int_{0}^{\infty}\frac{4\pi g}{r^2} dr
It is obvious that the integrand has a singularity at r = 0 and hence the integral (depending on 1/r) diverges. I could have demonstrated this without going through the math, but this gives some insight into why point charges are so important (and difficult to deal with classically) and not just man-made entities in a mathematical space.
Does this mean that the point charge is a source of infinite energy or stores infinite energy? The answer to this question in (old) classical electromagnetic theory (as given to students in school) is that the Coulomb field describing the point charge is singular at the location of the charge itself and the field is undefined so it does not make sense to go there; in
real life (and certainly on our planet) there is nothing like a point charge as charge must necessarily occupy some area and mass so the point charge model is self-contradictory and we might as well not worry about it too much. But isn't it interesting that a solid sphere (dielectric) composed of infinitely many infinitesimal sources of charge has a finite well defined electric field throughout its body even though we might be sitting on one of the "point" charges that compose it?
masudr has correctly pointed out that the charge density of a point charge is a delta function. Essentially a delta function centered at a point is an infinite spike at that point which suddenly falls off to zero as soon as you depart from that point. There is much more to the delta function and you can read all about it here:
http://en.wikipedia.org/wiki/Dirac_delta_function
The point of view given by prabhakar_misra is not correct. The fact that a point charge is capable of generating an infinite energy if it were to be "torn apart" somehow (refer to the integration above) is enough explanation for your (incorrect) statement that the law of conservation of energy will be violated: the electrostatic potential energy would be converted into kinetic energy of the fragments in your theory and there would be no violation.
But all this is handwaiving and the fact is that the classical point charge model being contradictory is no reason why we should ignore it. This handwaiving also shows that we mustn't stretch the classical theory too much having found out its weaknesses.
(Also, the stability of a charge distribution is described by Earnshaw's Theorem in the classical theory: a
collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges.)
Now to Saketh and quarks. Quarks are presumed to be building blocks of matter but in classical electromagnetic theory we assume at most that photons (or gauge bosons in the "advanced" terminology) mediate electromagnetic forces between charges--we do not dissect a charge to get down to quarks or other elementary particles that might possibly compose it. Besides, you will never see a lone quark--at least not in the next 24 hours
By the way, hadrons are not fundamental particles
(See http://www.mri.ernet.in/~sen/school.ps )
So if you are using Coulomb's law--which is a part of the Classical theory of Fields--you must accept the notion of a point charge as is using at most the Delta function in analysis. The contradictions in the model can be refined only using a modern picture of electromagnetic theory invoking also quantum theory (the refined picture is a Quantum Field Theory. QED is a quantum field theory for example). The divergence in classical theory can be attributed to the dependence of energy density of a point charge on an inverse power of radial distance r. You cannot do much about it until you switch camps to QFT
As for the uncertainty principle, Andrew has already provided the explanation: the distance (or any such physical quantity) can never be "exactly zero". Remember that in new quantum theories of nature (particles, fields, strings, whatever) determinism has given way to probability and nobody complains about non-deterministic theories anymore...in fact it is the starting point as predictions from QM have proved to be amazingly accurate in experiment. How can you even
know whether a physical quantity is exact in the first place? You must always accept a possible fluctuation (or noise) in a physical quantity--the uncertainty principle has (rightly) humbled us into submission.
Interestingly, Feynman and Wheeler had thought about the self-action of an electron on itself (and in general the self-action of a charge on itself). Their results came under what Feynman called Half Advanced and Half Retarded Potentials (also see Feynman Wheeler Theory:
http://en.wikipedia.org/wiki/Feynman-Wheeler_theory which ignores the self-action completely) but they were probably never published by this name.
The phenomenological theory of self-action certainly does not involve classical (coulombic) theory. So my last bit on this is that theoretically, a charged particle is capable of acting on itself but to even scratch the surface you must use methods from the quantum theory of electromagnetism. And I would be interested to learn how a self-action parameter can even be measured in the laboratory.
Saketh: For an overview of what String Theory
does and aims to do in
future you can start by reading
http://en.wikipedia.org/wiki/String_theory (nommathematical). But String Theory has a long way to go. The point charge has been dealt with in QFT/QED using a refined model.
(A search on wikipedia for self-action/self-force yields results on Self Defence
, so I have given up for now...I'll write more on this thread if I come across something interesting.)
