# Does a charge repell itself?

1. Jul 6, 2006

### hrishikesh

hmm here it goes
i know the electromagnetic (or whatever) force is equal to the product of two charges upon square of distances btn them (isnt it??)
so, if the distances betn them is 0, then the above eqn would be Infinite.
ie the force will be infinite and the charge will just explode?!? :grumpy:

i still havent deeply studied this topic but thats what i think.:surprised

So.. does a charge exert a force on itself? if it does.. please explain.

yo
-ђгเรђเкєรђ911:yuck:

Last edited: Jul 6, 2006
2. Jul 6, 2006

### masudr

The problem arises from the fact that a point charge is represented by a charge density function that is a delta function, which is, of course, unphysical.

3. Jul 6, 2006

### DaveC426913

Sure, that's how they reproduce! Mitosis!

4. Jul 6, 2006

### Andrew Mason

The distance between them is not the distance between the charge surfaces. While we speak of point charges, there is no such thing, really. The electron is the closest thing to a point charge but the uncertainty principle provides a limit to how accurately we can locate it. So the distances between charges can never be exactly 0.

This is a pretty good question. The answer is still not settled, as far as I can tell. http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html" [Broken] is a very entertaining and clear summary of Feynman's attempts to wrestle with this concept.

AM

Last edited by a moderator: May 2, 2017
5. Jul 7, 2006

### prabhakar_misra

if a charge were to repel itself that means it would experience a force due to its own field. which means it would start moving due to repulsion . which means it will develop energy without any external work. which is not possible by law of conservation of energy hence this is not possible .

even if the charge were to explode , its fragments will have kinetic energy, which doesn't have any source, thus again a violation of law of conservation of energy .

6. Jul 7, 2006

### Saketh

Like Andrew Mason said, point charges have no direct physical representation in the real world - they are merely a convenience for physical calculations.

I thought quarks were smaller particles with a smaller charge?

Each of the particles in a small charge (say, a charged golf ball) exert forces on each other. If we make the golf ball microscopic i.e. hadron-sized, the constituent parts inside the hadron exert forces on each other, but are confined by the strong force. A point charge would have no constituents, or so I think. If it did, then it wouldn't be a point charge anymore.

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.

7. Jul 9, 2006

### maverick280857

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 [Broken])

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 dependance 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 :tongue2: , so I have given up for now...I'll write more on this thread if I come across something interesting.)

Last edited by a moderator: May 2, 2017
8. Jul 9, 2006

### tehno

Why do you think the electron is supposed to explode?
It isn't called an elementar particle in vain.
It's so small that we don't know wether the laws of electrodynamics apply inside it,it must be a source and generator of its' own electromagnetic field.
If you think an analogy of a conductive sphere,than all the electricity is located on the surface .Meaning there's no need to think of any explosion.
But such view is too guessing.
We don't know the inside structure of the electron.
Many experts think the electron has no structure.
It's just the most fundamental "object" representing the free stable charge
in the free space.
On the contrary,the quarks can't exist alone but bounded in the nucleus and can have fraction of the unit charge (say 2/3 * 1,6*10^-19 As).
The question is over the top of my head,over the top of head of many scientists.
A.Einstein thought that mass of electron somehow arises from its' electromagnetics.He never got around to figure out that process (if there's such at all).
Too complicated,its' definitelly not a classical physics.

EDIT:
I would just mention the old saying of Torricelli:"The nature isn't affraid of emptiness".
But we KNOW today it IS affraid of infinities.
Electron is a perfect example of It's way of solving such problems.
The fundamental particle of a unit charge.You can't get smaller than that potential energy free...

Last edited: Jul 9, 2006
9. Jul 9, 2006

### vanesch

Staff Emeritus
I would like to point out that EVEN in QFT, the problem remains (of infinite self energy of the field), but the infinity can be dealt with somewhat nicer, by renormalization. Technically, the 1/r divergence in classical physics becomes a ln(r) divergence in QFT.
But it is still one of the most puzzling questions in electromagnetism. Another puzzling aspect, related to it, is the radiation reaction. An accelerated charge radiates EM energy (and hence there's a kind of "friction force" acting upon the radiating point charge). One can calculate this force through conservation of energy, and then introduce it by hand, but it is not obvious how to deduce it directly from electromagnetism. Nevertheless, all synchrotrons work on that principle.

10. Jul 9, 2006

### maverick280857

Yes I was referring to renormalization without referring to it explicitly. My point was to state that QFT at least gives a way of dealing with the problem and that is better than $\infty$ staring in our face in the classical theory

Yup...it sure is interesting (its one of the things in physics which really hits me in the eye when I read about it and its always a source of new things to learn). By the way what does string theory have to say about these issues? Or other quantum field theories? Perhaps we should have a separate thread about these issues.

Last edited: Jul 9, 2006
11. Jul 9, 2006

### tehno

As of now,nothing fundamentally new.
There is quantization of the fields ,and there is mathematical formalism how the sources of these fields are treated (ie particles ,bozon mediators etc.)AFAIK,the thinking of the most is that the most beautiful mathematical theory is the closest to the ultimate truth .
But nothing can guarantee that such view must be always true.
STOP.

Last edited: Jul 9, 2006
12. Jul 9, 2006

### masudr

But classical electromagnetism is just such a cool theory otherwise. What do we think? Spacetime must be quantized?

13. Jul 10, 2006

### Farsight

I think this is the time to ask What is Charge?

It's a fundamental property, a parameter, a measure, a dimension, and I personally look to topology to explain it and the forces that betray its presence. If topology is the best place to look, it doesn't make sense to talk about a distortion/slope/fold/loop/knot of zero height/width/length. So "Why doesn't a point charge explode?" doesn't make sense either.

14. Jul 11, 2006

### maverick280857

Okay so technically speaking then, the divergence still exists in QFT so I wasn't right in saying that it is "better" in QFT (except possibly if you resort to renormalization, which I do not know much about yet).

15. Jul 12, 2006

### maverick280857

In much the same way as Dirac Delta fxn can be constructed from a sequence of functions or rectangles with unit area whose width thends to zero and height to $\infty$ can't a distortion/slope/fold/loop/knot of zero height/width/length be described as the limiting case of similar topological entities with nonzero dimensions?

I am sorry I do not know much about topology so this might be a "weird" question to say the least.

16. Jul 14, 2006

### Farsight

I don't know maverick. What I was trying to say is something like: you can have some simple topological intangible thing like a crease. This has zero width, and zero height. But it can't have zero length as well. There has to be some non-zero dimension somewhere, otherwise the topological "thing" wouldn't be what it is. In general you can have a point in topology, such as the apex of a cone, and this might be what you experience or "see", but I don't think you can have a point all on its own.

17. Jul 14, 2006

### pinestone

ker-boom

Well maybe it already exploded. Instead of the force moving away, it just stays in "orbit"...like the arms of a spiral galaxy.

Last edited: Jul 14, 2006