Does a Point Charge in Classical Electromagnetism Possess Self Energy?

[Meir Achuz]

TL;DR

This shows thata point charge in classical electromagnetism has no 'self energy'.

This PDF shows that a point charge in classical electromagnetism has no 'self energy'.

 

[renormalize]

This PDF shows that a point charge in classical electromagnetism has no 'self energy'.

In your attachment you assert:

This is certainly true for a point "test charge" with infinitesimal $q\,$, which by assumption does not contribute to the potential $\phi$. But can you cite a textbook reference that justifies and agrees with your last two sentences above for a point charge in the case where $q$ is finite?

 

[Meir Achuz]

"Chapter2 Electrostatics so,ifyouhavesetthereferencepointatinfinity, W=QV(r)."
You added the word "infinitesimal".
"W= 1/ 8π\\epsilon_0 sum n i=1 n j=i qiqj/rij (2.41) (wemuststillavoidi=j,ofcourse)."

 

[renormalize]

"Chapter2 Electrostatics so,ifyouhavesetthereferencepointatinfinity, W=QV(r)."
You added the word "infinitesimal".
"W= 1/ 8π\\epsilon_0 sum n i=1 n j=i qiqj/rij (2.41) (wemuststillavoidi=j,ofcourse)."

Can you please clarify the citation by providing the author, title and page number of the text where this quotation appears? Thanks!

 

[Meir Achuz]

I thought you would recognize Griffiths. Actually, the shoe is on the other foot. Can you cite a textbook reference that justifies and agrees for an electron in the case where its charge is infinitesimal??

 

[renormalize]

Actually, the shoe is on the other foot. Can you cite a textbook reference that justifies and agrees for an electron in the case where its charge is infinitesimal??

Absolutely. From Jackson, Classical Electrodynamics 2nd ed. pg. 28 (emphasis added by me):

So by definition, the electrostatic field $\vec{E}$ in eq.(1.1) ignores the effect of the charge $q$ on itself, but only in the limit $q\rightarrow 0$ (i.e., infinitesimal charge). This same restriction holds for the relation between the work and the potential $U$:$$\vec{E}=-\nabla U\Rightarrow\vec{F}=-q\nabla U\Rightarrow\text{work}\equiv\int\vec{F}\cdot d\vec{r}=-q\int\nabla U\cdot d\vec{r}=-q\,U+\text{const.}$$On the other hand, for $q$ a finite charge, $U$ will contain a contribution from the charge's own (self) $\vec{E}$-field, which is indeed divergent if that charge is concentrated at a point.

 

[Meir Achuz]

Don't ignore the footnote, which I have attached. Jackson points out that infinitesimal charge is mathematical and NOT PHYSICALLY POSSIBLE, The point here is that to define E, even a small charge can polarize the source of E, making the force equation bilinear. It is completely different for energy where Jackson knows the charge cannot be infinitesimal. (See Section I.1. See also pages 40,41, which is more like Griffiths. Jackson uses i<j. I see my picture left out "impossible to"

 

[renormalize]

See also pages 40,41, which is more like Griffiths. Jackson use i<j

What Jackson says there regarding discrete charge distributions is (my emphasis added):

So the self-energy terms are there in principle but are dropped by convention. This is allowed because they are (infinite) constants that simply set the zero of the energy $W$ and are thus ignorable.
That situation changes when we consider continuous charge distributions:

because the integrand in (1.52) includes the points $x=x^{\prime}$; i.e., self-energy is inevitably included in the double integral that defines $W$.
Feynman comments on how this affects the energy of a point charge in his lecture https://www.feynmanlectures.caltech.edu/II_08.html:

So in classical electrodynamics it seems to be unavoidable that infinite self-energy must accompany any truly point charge.

 

[Meir Achuz]

This is getting silly. Neither of us will be convinced, but you keep putting up half truths, half wrong.
I will try again. You neglect what you don't want. Jackson's equations 1.47, 1.48, 1.49, 1..50 clearly show there is no i=j term. For some reason, he says it in words for Eq,. (.51). You, not he, adds, "So the self-energy terms are there in principle but are dropped by convention." His equations clearly show that it is not there by absence, not "convention". You made up your "convention". When you quote someone don't make up what they didn't say.
My original post was limited to discrete charges. Do you find anything wrong there? I will respond to the question of going from discrete to integrals in another post.

Reference: https://www.physicsforums.com/threads/self-energy.1083048/

 

[A.T.]

Jackson's equations 1.47, 1.48, 1.49, 1..50 clearly show there is no i=j term. For some reason, he says it in words for Eq,. (.51).

Not for "some reason", but simply because in equation 1.51 (as it's written) there is a i=j term, so he needs to specify that it should be ignored. Whether you call that a "convention" or not doesn't really matter.

 

[renormalize]

My original post was limited to discrete charges.

Yet the very first sentence of your original post was:

TL;DR: This shows thata point charge in classical electromagnetism has no 'self energy'.

which contradicts the Feynman lecture I quoted and is clearly wrong.
Are you espousing a personal theory?

 

[Meir Achuz]

Of course, I mistyped. It should have been, it IS there by absence.

Of course, I mistyped. It should have been, it IS there by absence.

Yet the very first sentence of your original post was:

which contradicts the Feynman lecture I quoted and is clearly wrong.
Are you espousing a personal theory?

I did write, "This PDF shows that a point charge in classical electromagnetism has no 'self energy'."
That's because there is no self energy term in the sum of charges. Thank you for pointing it out.
Since the continuous integrals are derived from the sums, I thought that was enough.
I didn't know I was contradicting Feynman's lecture. I will add that to my resume.
I have to discontinue my replies. I was planning to patent a car that runs on the infinite self energy of electrons, with no other fuel needed, but I have already been scooped:

 

[A.T.]

I was planning to patent a car that runs on the infinite self energy of electrons,

To release that self energy you would have to split an electron in two halves that repulse each other. Tricky!

 

[Matthias_Rost]

Guys you are up to Einsteins last Question: "What is an electron?" And to answer that, you'd first need to answer: "What is Charge?" (Not how much is elementary charge q) "What is mass?" and: "What is Spin?"

See we can all agree, that Electrons are metastable particles, which have a minimum of charge, which is called the elementary charge.
Irrespective of any Equations you use, it has proven first by Millikan that there is no charge smaller than q. Since in the Millikan-Experiment, you get discrete lines of charged oil particles of an ensemble of oil particles.
If there were infinitesimal charge amounts, then you would get a continuous spectrum and no discrete one.
Plus if you question the existence of a minimum charge, then you also question the Atom-Theory of Democrit. And it took over 2000 years to prove it. Which was done by Ernest Rutherford and the Experiment is nowadays called RBS. So no your assumption of infinitesimal small charges is pure nonsense. Mathematically feasible, but it has nothing to do with physics and therefore reality.


So in principle, we can also agree, that most mass in Atoms is binding energy of the corresponding particles, except for self energy and the proposed Higgs-Mechanism, do we? When we assume a single electron, without any interaction, it still has to have a spin, which is an intrinsic rotational momentum. This is where I would agree to the original statement of the post. If there was an infinitesimal small amount of charge, it had to be bound in a particle, which one? Because same goes for the spin of electrons. If you reduce the radius to 0 you would get infinitesimal high rotational momentum, which makes no sense. Plus an intrinsic rotational momentum has a direction, which is contradictious to the belief, that electron are spheres. Because spheres are isotropic objects and spin is anisotropic. So, how can an isotropic object have an intrinsic anisotropic property. Makes no sense. The same goes for the assumption that electrons would be point objects. Here the radius is zero, but the object is still undirected and therefore isotropic.

And this was the original mindboggling thought-experiment of Einstein.

So rather than copy and paste what there is already, you should consider other geometries than 0-D and 3-D.

And to assume that an Equation predicts reality is mindboggling wrong.
Equations are there to describe observations. And Coulombs law, irrespective of how it's written, is always proportional to the distance of two charged objects. When the distance is zero, the self energy would be infinitesimal high, which contradicts energy conservation.

I hate how physics, which is a greek word and means "condition of reality" is reduced by mathematical formalism into absurdum.

Plus you'd have to check out the annihilation process. When an electron and a positron collide they produce photons. Which makes electrons and positrons metastable. And there is no particle known which has a smaller charge than q. Protons, electrons, positrons and Electons all have the charge of |q|. And you can add charge up, but how to divide a charge carrier into multiple particles? Show me and I believe you.

But wait, that is exactly the annihilation process. So let's reverse the process of annihilation in a thought experiment. "How can two Photons of 0,511 MeV generate two charge carriers?" And when you found out how, then you know, what an electron is, what charge is and what spin is. Seriously.

And all this dispute about mathematical formalism is silly. Definition of science: "Two truthful statements can't contradict each other." And all you do is to rewrite a formula in different ways. And then you ignore the details expressed by each other, and then you state, how is it written in the text book? That is Academia and no Science.

What is this? Why don't you derive the Coulomb Law from scratch and then confirm it with experimental data. This would be the scientific way.

 

[Meir Achuz]

To release that self energy you would have to split an electron in two halves that repulse each other. Tricky!

It was a joke, but read the insert.

Guys you are up to Einsteins last Question: "What is an electron?" And to answer that, you'd first need to answer: "What is Charge?" (Not how much is elementary charge q) "What is mass?" and: "What is Spin?"

See we can all agree, that Electrons are metastable particles, which have a minimum of charge, which is called the elementary charge.
Irrespective of any Equations you use, it has proven first by Millikan that there is no charge smaller than q. Since in the Millikan-Experiment, you get discrete lines of charged oil particles of an ensemble of oil particles.
If there were infinitesimal charge amounts, then you would get a continuous spectrum and no discrete one.
Plus if you question the existence of a minimum charge, then you also question the Atom-Theory of Democrit. And it took over 2000 years to prove it. Which was done by Ernest Rutherford and the Experiment is nowadays called RBS. So no your assumption of infinitesimal small charges is pure nonsense. Mathematically feasible, but it has nothing to do with physics and therefore reality.


So in principle, we can also agree, that most mass in Atoms is binding energy of the corresponding particles, except for self energy and the proposed Higgs-Mechanism, do we? When we assume a single electron, without any interaction, it still has to have a spin, which is an intrinsic rotational momentum. This is where I would agree to the original statement of the post. If there was an infinitesimal small amount of charge, it had to be bound in a particle, which one? Because same goes for the spin of electrons. If you reduce the radius to 0 you would get infinitesimal high rotational momentum, which makes no sense. Plus an intrinsic rotational momentum has a direction, which is contradictious to the belief, that electron are spheres. Because spheres are isotropic objects and spin is anisotropic. So, how can an isotropic object have an intrinsic anisotropic property. Makes no sense. The same goes for the assumption that electrons would be point objects. Here the radius is zero, but the object is still undirected and therefore isotropic.

And this was the original mindboggling thought-experiment of Einstein.



And to assume that an Equation predicts reality is mindboggling wrong.
Equations are there to describe observations. And Coulombs law, irrespective of how it's written, is always proportional to the distance of two charged objects. When the distance is zero, the self energy would be infinitesimal high, which contradicts energy conservation.

I hate how physics, which is a greek word and means "condition of reality" is reduced by mathematical formalism into absurdum.

Plus you'd have to check out the annihilation process. When an electron and a positron collide they produce photons. Which makes electrons and positrons metastable. And there is no particle known which has a smaller charge than q. Protons, electrons, positrons and Electons all have the charge of |q|. And you can add charge up, but how to divide a charge carrier into multiple particles? Show me and I believe you.

But wait, that is exactly the annihilation process. So let's reverse the process of annihilation in a thought experiment. "How can two Photons of 0,511 MeV generate two charge carriers?" And when you found out how, then you know, what an electron is, what charge is and what spin is. Seriously.

And all this dispute about mathematical formalism is silly. Definition of science: "Two truthful statements can't contradict each other." And all you do is to rewrite a formula in different ways. And then you ignore the details expressed by each other, and then you state, how is it written in the text book? That is Academia and no Science.

What is this? Why don't you derive the Coulomb Law from scratch and then confirm it with experimental data. This would be the scientific way.

I appreciate that you have expended time and knowledge on my post, which I had thought was straightforward. I certainly didn't mean to bring Einstein into it.
We have some agreement, but I will comment on some things:
1. You wrote,"So in principle, we can also agree, that most mass in Atoms is binding energy of the corresponding particles, except for self energy. "self energy" is a vague term here. If you mean intrinsic mass and, thus rest energy, I agree, but my view is that it is not 'electromagnetic self energy'.
2. You wrote, "Electrons are metastable particles, which have a minimum of charge", and explain what you mean. If electrons are metastable, then no particle is stable (except, possibly, 'hidden matter', which then will never be observed.) Also, I prefer that they have an exact charge.
3. After demonstrating that a point particle with an intrinsic magnet moment has a non-isotropic magnetic field, you wrote, "Here the radius is zero, but the object is still undirected and therefore isotropic.." It's not the 'shape' of a point, but the shape of its field that is relevant.
4. You wrote, "the original mindboggling thought-experiment of Einstein." I don't know which 'mindboggling' thought-experiment you mean. Anyway, as a physicist, I believe in real experiments, and not 'mindboggling' thought-experiment. ALL the mistaken paradoxes of SR are based on mindboggling thought-experiments.
5. "Which makes electrons and positrons metastable."
You seem to be saying that ANY interacting particle is 'metastable. I thought that particle that decays in isolation is metastable.
6. "I hate how physics, which is a greek word and means "condition of reality" is reduced by mathematical formalism into absurdum." That's how Einstein described Minkowski until he needed him.
7. I'm afraid I don't understand your last three paragraphs.

With all you have written, do you see a mistake in my original post?

 

[Matthias_Rost]

It was a joke, but read the insert. --> He who takes jokes seriously didn't understand them.
And if e+ + e- collide they do form two photons, which traject in opposite directions. The Self energy of one electron/positron is 0,511 MeV. And both Photons have an energy of 0,511 MeV. So when it was j joke, where's the point?

1.
See the law you apply is Coulomb and Coulomb is electrical energy. If you'd account also for weak and strong force, please describe it.
2.
Yeah sure. q is the discrete minimum Charge. And yeah there was a long time we thought atoms were stable. Today we know, that iron is the most stable core. But if you raise the energy density high enough you can even split an iron atom core. So what's the difference between stable and metastable. The use case I would say.
3. After demonstrating that a point particle with an intrinsic magnet moment has a non-isotropic magnetic field, you wrote.
NOPE!! I've never proven, that an electron has an intrinsic magnetic moment, never ever, it has a spin. What's the difference between spin and magnetic moment? Small Tip, there no magnetic monopoles. "Here the radius is zero, but the object is still undirected and therefore isotropic.." It's not the 'shape' of a point, but the shape of its field that is relevant. --> did I stutter: How can a point, so a zero dimensional object has a field, which has a radius and the field is rotating in one direction. a point, doesnt have any intrinsic preferred direction. So if you state: "there's a point (or maybe sphere) which is isotropic but has an unisotropic field you violate the law of causality. How in the world and please begin a sphere. A Spin can simply be displayed as a vector. So in which direction, should this vector go? Doesn't make any sense since a point or a sphere doesn't have any preferred (anisotropic direction), therefore your statement is acausal and therefore nonsense.
4. You wrote, "the original mindboggling thought-experiment of Einstein." I don't know which 'mindboggling' thought-experiment you mean. Anyway, as a physicist, I believe in real experiments, and not 'mindboggling' thought-experiment. ALL the mistaken paradoxes of SR are based on mindboggling thought-experiments.

Ahhh you believe. A belief is simply in opinion. See every real experiment you can check out has shown, that the measurable radius of an electron, assuming it to be a sphere, is smaller than the detection limit. So everything we proven yet is. Either the sphere is way smaller than the detection limit or 3-D Geometry simply doesn't apply. And the same goes for a dot, since a dot, would be simply a sphere with a radius of zero. So in principle, nothing has change, except that you closed your eyes and belief, that the is just smaller than the detection limit.
5. "Which makes electrons and positrons metastable."
You seem to be saying that ANY interacting particle is 'metastable. I thought that particle that decays in isolation is metastable.
And again, if a particle or substance is metastable depends on the application case. See Water melts at 0°C and it boils at 100 °C but only at 1 bar pressure. We nowadays can even split atoms in nuclear plants, although atom is greek and means undivisible, which means nothing else than stable. Yeah and it most cases most atoms are stable. And then some cases they're not. Same goes for electrons. There's even a measurement principle called positron annihilation spectroscopy (lifetime spectroscopy) with which you can measure the lifetime of positron entering a material and destroying electrons in the material, by detection of the resulting radiation. Which is mostly used to analyze ceolithes, which are mostly used for self cooling beer barrels, that you can buy in your supermarket. So to make sure. I'm talking about real physics and no mumbo jumbo.
6. "I hate how physics, which is a greek word and means "condition of reality" is reduced by mathematical formalism into absurdum." That's how Einstein described Minkowski until he needed him.
Who said, that Einstein finished the ART and the SRT? He said himself, that it's not sufficient, regardless of Murkowski contributions.
7. I'm afraid I don't understand your last three paragraphs.
What is a thought experiment? You have to imagine something yourself. So imagine you have two Photons of 0,511 Mev and they approach each other in opposite direction and nearly "collide". What happens? If you have any idea of how an e- + e+ can form that way, you're on the right track.

 

[berkeman]

It was a joke,

What was a joke? And why are you trying to make obscure jokes in a technical discussion thread?

 

[Meir Achuz]

I apologize. I shouldn't have used that word in a technical discussion thread. The point was to show by the insert where taking 'self energy' too seriously can lead.

 

[Meir Achuz]

an intrinsic magnet moment has a non-isotropic magnetic field, you wrote.

Thanks for your interest, but I guess we are at an impasse.
The quote was something like, “with all the new math being applied to my theory, I hardly understand it any more myself.”

 

[Matthias_Rost]

Thanks for your interest, but I guess we are at an impasse.
The quote was something like, “with all the new math being applied to my theory, I hardly understand it any more myself.”

Yeah, then the better way would be to stick to mathematical formulations, which are easy to understand. Because writing papers with absurdly and highly complicated math's has nothing to do anymore with science. Rather than academia. And Academia is boring, as you might know.

So from my point of view everything that is based upon matter has to have a self energy, since Matter is simply one form of Energy, at least according to E=mc².
There's even a good joke about it?

There's even a good quote on that topic by a friend of mine from Simbabwe.

 

[weirdoguy]

since Matter is simply one form of Energy

This is simply wrong. Energy is a property of matter, one of many.