Talk:Classical electron radius

Link

Please note http://atdotde.blogspot.com/2005/09/natural-scales.html --Pjacobi 20:33, 15 October 2005 (UTC)

Good article

This is a smashing good little article. Clear, consice and to the point. Good work all! Kevmitch 04:11, 19 April 2006 (UTC)

Exact energy of charged sphere

Isn't the formula given the energy required to assemble a charged sphere with all the charge on the surface? If it's constant charge density, I believe a prefixed factor of 3/5 is needed.

Actually the correction factor of 3/5's is for a charged sphere with all the charge uniformly distributed over the volume, not over the surface.

So for a uniform volume distribution you should get

U = 3 / 20 * whatever

and for a surface charge distribution you get (if I remember correctly):

U = 1 / 8 * whatever.

-- Eugene Y.

Thats where "approximately" comes in. For the energy E of a sphere of radius R containing a constant charge density I get:

$E={\frac {3}{5}}\,{\frac {e^{2}}{4\pi \epsilon _{0}R}}$

and for the energy of a sphere of radius R containing a constant surface density I get:

$E={\frac {1}{2}}\,{\frac {e^{2}}{4\pi \epsilon _{0}R}}$

I have changed the text to be more exact. PAR 14:58, 2 November 2006 (UTC)

electron radius

if the electron travels fast enough to affect its mass, would the radius of the electron be smaller or larger???

Counter-intuitively, it gets smaller. See Discussion section in Article.

question about derivation

By request, I replaced the original derivation with one based on energy-conservation. Both approaches yield the same answer, but the derivation based on energy-conservation seems easier to follow, and answers many of the questions raised. The discussion below relates to the original derivation. PBWilson, 10-3-2016 — Preceding unsigned comment added by 184.98.190.165 (talk) 23:44, 3 October 2016 (UTC)

Thanks very much for the article, but I was wondering how you derived the constants in front of the expressions (1/2 or 3/5)? I have tried integrating 1/2*[epsilon]*[E field]^2 dr three times, and get the basic term but not the constant. Nor can I find this on any other website (they all seem to ignore the constant). Please could you add the derivation to the article? I'd really appreciate it - thanks for your help! P.S. why does everyone, even this article, ignore the constants when quoting the radius? Thinkingabout physics (talk) 21:44, 16 April 2008 (UTC)

I don't know where your expression 1/2*[epsilon]*[E field]^2 dr came from, but the way I did it was to realize that bringing in an infinitesimal charge dq from infinity to the surface of a ball of radius r having constant charge density ρ requires the same amount of energy as bringing dq from infinity to a distance r from a point charge q where

$q={\frac {4}{3}}\pi r^{3}\rho$

The energy needed to do that is

$dU={\frac {1}{4\pi \epsilon _{0}\ r}}qdq$

So, to build a shell of thickness dr with the same density on that ball requires bringing a charge $dq=4\pi r^{2}\rho \,dr$ from infinity. (This assumes that the charge on the shell is so small that interactions between charges on the shell add negligible energy). That needs energy

$dU={\frac {1}{4\pi \epsilon _{0}}}\,\,\left({\frac {4}{3}}\pi r^{3}\rho \right)\left(4\pi r^{2}\rho \,dr\right)$

To build a ball from scratch, you integrate from 0 to R, the radius of the ball, remembering that ρ is constant, you get

$U={\frac {1}{4\pi \epsilon _{0}}}{\frac {(4\pi \rho )^{2}R^{5}}{15}}$

replacing ρ with the charge divided by the volume ($4\pi R^{3}/3$) gives

$U={\frac {3}{5}}\,{\frac {1}{4\pi \epsilon _{0}}}{\frac {q^{2}}{R}}$

The reason people ignore the constants is because the classical derivation is not valid at the scale of the electron, its just a hand-waving argument, but it does yield a constant with dimensions of length which is relevant in quantum mechanics as well. The constant multiplier loses significance when going to quantum mechanics, but the function of fundamental constants does not. PLEASE NOTE - the above derivation needs to be checked! Yes, I know, its inconceivable that I would make a mistake, but it has been known to happen on rare occasions. PAR (talk) 15:50, 18 April 2008 (UTC)

I just did the dimensional analysis and you must have made a mistake. Two, in fact, which happened to compensate each other. First you forgot to do the integral in the expression for dU (= W_shell) and then you mistook the power of r before integration. 82.139.87.95 (talk) 07:59, 23 September 2009 (UTC)

I agree there is an error in expression for $dU$ from dimensional point of view. It should be $r$ in devisor. Otherwise the integration and the result is correct.— Preceding unsigned comment added by AlexKritov (talk • contribs) 05:21, 17 November 2019 (UTC)

Density

Given a (classical) sphere with a radius of 2.818×10⁻¹⁵ m, this yields a spherical volume of 9.378×10⁻⁴⁴ m³. Combined with a mass of 9.109×10⁻³¹ kg, this gives a matter density of 9.718×10⁹ g/cm³. This is about 30 times greater than the approximate density of an atomic nucleus of 3×10⁸ g/cm³. — Loadmaster (talk) 22:52, 9 March 2011 (UTC)

The mass of the electron in the theory of classical radius is all in the field on the outside of the sphere. The total mass inside the sphere must be zero. — Preceding unsigned comment added by 2003:C9:8BDE:3BF8:4AE3:6DDC:2CE9:A3A4 (talk) 17:31, 22 December 2017 (UTC)

Clarity

I would estimate that the average non-math-savvy user will arrive at this page due to a search for the radius/diameter of an electron, and expect to --somewhere on the page-- see the answer in plain english. I would suggest providing it. SentientSystem (talk) 04:27, 22 September 2011 (UTC)

Plain English. No one actually knows, because no one is sure if the electron has internal structure or not, as is the case with all elementary particles. Frankly I find it quite odd that a proton has internal structure which is the cause of it's permanent electric and magnetic dipole moment and charge, while we consider an electron not to have internal structure, yet still posses a permanent electric and magnetic dipole moment and charge. When both electric and magnetic dipole moments are quite impossible without a system of charges moving internally in relation to one another. "In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of electric charges, that is, a measure of the charge system's overall polarity." See Electric_dipole_moment. "The electron is a charged particle of charge (−1e), where e is the unit of elementary charge. Its angular momentum comes from two types of rotation: spin and orbital motion. From classical electrodynamics, a rotating electrically charged body creates a magnetic dipole with magnetic poles of equal magnitude but opposite polarity." See Electron_magnetic_dipole_moment

So, if it has a uniform electric charge, then the rotation of that uniform charge cannot create two oppositely charged poles. Magnetic monopoles do not exist. It takes a minimum of two charges to create a magnetic dipole, two opposite charges or in reality two charges of differing potential. It's the only known laboratory tested way. "The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. Consider a bar magnet which has magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strength p of its poles (the magnetic pole strength), and the vector ℓ separating them." See Magnetic_moment

In other words it must have internal structure composed of both negative and positive charges, or more accurately, charges of higher and lower potential.84.226.185.221 (talk) 11:00, 12 October 2015 (UTC)

Planck's Constant Conflict

The very final point in this article states "$\hbar$ is Planck's Constant", isn't $\hbar$ the reduced plancks constant (i.e h/2pi)? Anonymous capybara (talk) 13:38, 2 December 2013 (UTC)

Only if there is chance of confusion. It is only because people like to use wavelength instead of wavenumber that we have this problem. In any case, it is more usual to ignore the reduced part when using words, as long as the equations are right. Gah4 (talk) 10:19, 13 March 2021 (UTC)

Classical radius of proton

The article Thompson scattering says that a quantity called classical radius of a charged particle is convenient to be defined. Can also proton have a classical radius defined for for it?--5.2.200.163 (talk) 13:49, 12 February 2016 (UTC)

I suppose so. The classical electron radius assumes that all the mass is due to the energy in the electric field. Protons have actual mass in addition to any electric field mass. It gets more interesting for the neutron. Note that a neutron also has a magnetic moment, and even has a charge distribution. Gah4 (talk) 07:16, 13 February 2022 (UTC)

This should be made much more simple...

Introducing the classical electron radius in terms of the interaction of positrons and electrons is needlessly obscure and historically inaccurate. The standard, straightforward discussion of where it comes from is in terms of the electrostatic self-energy of a charged sphere. No need to mention the virial theorem, radiative energy loss, and other complexities. Keep it as simple as reasonably possible.

Electron (rest) mass

Qwerty123uiop, I am curious. In your edit comment ("Remove recently-added obsolescent reference to "rest" mass. Mass is a Lorentz invariant. There electron mass is what it is, not just in frames where it is at rest."), it takes me time to figure out that you appear to think that every reader will understand that the rest mass and not the relativistic mass is meant. A particle's rest mass is Lorentz-invariant. Its relativistic mass is a component of the energy–momentum 4-vector. Particle physicists sometimes define the term "mass" to always mean "rest mass", but to others, this is silly. I'm afraid your inappropriate use of the term "obsolescent" does not help. Do you mean "obsolete" (which has a different meaning)? Do you claim that "rest mass" is an obsolete term (particularly in the broader context beyond particle physics, such that most readers will understand what is meant)? —Quondum 16:10, 13 December 2018 (UTC)

Hello Quondum. In my comment I used the word "obsolescent" to mean something that is on its way to being obsolete, but may not be there yet. You are mistaken to say that the relativistic mass is a component of the energy-momentum 4-vector. The energy-momentnum 4-vector is (E, p_xc, p_yc, p_zc), where E is the total energy and the pc's are the 3 components of the relativistic momentum. The invariant mass comes from the Einstein relation: mc² = sqrt( E² - ((p_xc)² + (p_yc)² + (p_zc)²)). If you know 4 of those 5 quantities you can get the 5th. The Lorentz gamma factor then arises in several ways, including gamma = E / mc². This is spelled out in some detail in the article on electron rest mass and more so in the article on mass in special relativity.

In a Wikipedia article like this it is entirely appropriate to explain, up front, the different usages of terms that may be ambiguous. That is why I added an explanatory paragraph in electron rest mass but am not in favor of removing the term "rest" there. And do not worry about "most readers" understanding this. "Most readers", starting with beginning learners of SR, will understand the distinction. The classical electron radius article does not need to add to the confusion, hence we should leave the "rest" out there.

Qwerty123uiop (talk) 17:31, 13 December 2018 (UTC)

Hmm. Rewriting history with your version of Einstein's relation? I need no lessons from you on special relativity. And for a factor c² (which is at times by convention ignored), you call me mistaken. Anyhow, my curiosity is satisfied. —Quondum 02:06, 14 December 2018 (UTC)

Unsupported claim

Grufo, you have made the unorthodox claim

"If they got closer to each other than the classical electron radius the kinetic energy that they would have acquired would surpass their rest mass. But we do know that when an electron and a positron merge they annihilate producing two photons with the exact energy equivalent of the electron rest mass (511 KeV). We must therefore deduce that the annihilation took place not closer than the classical electron radius (more specifically, it took place exactly at the distance of 1 r_e in this specific case)."

You added three references, two of which do not support this claim (they use the classical electron radius in the formula for annihilation probability) and the third of which I do not have access to. You may chose to quote a passage from the third, but in my view you are adding original research that is furthermore non-accepted theory. —Quondum 23:07, 13 May 2020 (UTC)

@Quondum: What is unsupported? The sentence “the annihilation took place not closer than the classical electron radius”? Make them annihilate further apart and they will emit two photons less energetic than the two 511 KeV photons we get, make them annihilate closer and the two photons will have more energy than the original mass of the two electrons. That is because the classical electron radius is derived by definition as the point where the electromagnetic self-energy completely accounts for the mass. It doesn't mean that the electron has such radius, it can happily be a point-like particle, but it does mean that an electron and a positron cannot approach each other closer than the classical electron radius, or they will acquire too much kinetic energy (by definition of what the classical electron radius is). You can calculate yourself the energy released by an electron and a positron hypothetically merging at a distance $d$ equal to ${\frac {r_{\text{e}}}{10}}$ (i.e., one tenth of the classical electron radius) – electron charge and mass are all you need. --Grufo (talk) 23:53, 13 May 2020 (UTC)

Hello Grufo: what you are saying is not correct, sorry. The thought experiment of having two particles start at infinity is not relevant. If you let an electron annihilate with a positron at finite kinetic energy, the photons in the final state will have more energy than the mass energy of the incoming particles.... by exactly the amount of the excess kinetic energy. In your thought experiment the particles already have their mass energy at infinite distance. The extra (kinetic) energy they acquire via Coulomb attraction does not vanish after they annihilate - it simply gets added to the 0.511 MeV of energy that the final state photon leave with. There is no connection here to the classical electron radius. Qwerty123uiop (talk) 00:23, 14 May 2020 (UTC)

@Qwerty123uiop: Technically this is not correct. Imagine to give both the electron and the positron an energy of 1 MeV (1 MeV to each of them). They will merge releasing two photons of 1.511 MeV each. Now the question is simply: at which distance do a 1 MeV electron and a 1 MeV positron have to merge for releasing 1.511 MeV of energy each? And again the answer is: at a distance equal to $r_{\text{e}}$. Make them merge further apart and they will release less than 1.511 MeV each, make them merge closer and they will release more than 1.511 MeV each. This is because $r_{\text{e}}$ is the exact point where the $\Delta E$ of energy accumulated by the electrostatic attraction during the collision equals 511 KeV (i.e. the electron rest mass). Any energy surplus that you have given to the two particles gets only added to the final result, without changing the fact that they have to approach $r_{\text{e}}$ for accumulating the 511 KeV of kinetic energy that contributes to the final photons' energy. I repeat with different words what I have already said to Quondum: the classical electron radius is not a physical sphere and the electron can happily be a point-like particle, but the significance of the classical electron radius is inherently related to the amount of mass that an electron possesses. --Grufo (talk) 00:45, 14 May 2020 (UTC)

Hang on, both of you. Correctness/truth is not relevant in editing WP. Insertion of claims unsupported by references is not permitted. Grufo made a claim about distance apart of annihilation ("... the annihilation took place not closer than …"), which is WP:OR without being mentioned in any of the references. Any logic you choose to apply, correct or not, without a direct confirmation from the source, is simply irrelevant. —Quondum 01:51, 14 May 2020 (UTC)

@Quondum: I agree with the general rule, although with equations there is not much to quote. The formula for calculating the kinetic energy acquired by one of the two components of the pair (let's say the electron) in function of the distance where the merge happens is

$E=k_{\text{e}}{\frac {e^{2}}{d}}$

where

$E$ is the final energy released

$k_{\text{e}}$ is the Coulomb constant (8.9875517923×10⁹ kg⋅m³⋅s⁻⁴⋅A⁻²)

$e$ is the electron charge (1.602176634×10⁻¹⁹ C)

$d$ is the distance chosen by you as the radius where you would like the merge to happen

And indeed by replacing d with the classical electron radius r_e we obtain that each of the photons released has an energy of 8.187105644 × 10⁻¹⁴ J (equivalent to the electron's rest mass).

What would happen if instead of r_e we imagined the merge happening at different distances? Let's see…

Arbitrary merge distances and energies released

Distance chosen (d)	d / re	Energy released (E)
2.8179403227 × 10−16 m	1/10	8.187105644 × 10−13 J/c2
1.408970161 × 10−15 m	1/2	1.637421129 × 10−13 J/c2
2.8179403227 × 10−15 m	1	8.187105644 × 10−14 J/c2 (electron rest mass)
5.635880645 × 10−15 m	2	4.093552822 × 10−14 J/c2
2.8179403227 × 10−14 m	10	8.187105644 × 10−15 J/c2

All this is obvious, because the classical electron radius was created with the precise purpose of fitting in that equation and give the electron rest mass as a result (because that was what they could see in the experiments: photons of 511 KeV). This does not say anything about what really happens during the annihilation. It might be that the electron and the positron keep approaching further until they touch the Planck length and transform in a polar bear and a dinosaur. That said, the equation above does tell us though that our understanding stops at r_e. And whatever happens afterwards is a fairy tale, because we don't have any math for it – and the truth is that the math that we have tells us that we would obtain more mass than the electron mass if the two approached further. --Grufo (talk) 03:02, 14 May 2020 (UTC)

@Quondum: (I quote from the History) “The extensive beating about the bush while refusing to acknowledge this simply being obstructive” I see the attitude here… However I have limited myself and I have restored only the part with obvious and incontestable references. Working on the rest. --Grufo (talk) 01:12, 15 May 2020 (UTC)

The electron's mass-to-charge ratio was measured in the 1890s, and after Millikan's oil-drop experiment, the mass could be calculated; so, the electron mass was known by 1909, long before the discovery of positrons in 1932. The earliest calculations of a "classical electron radius" of which I am aware were based not on electron-positron annihilation or trying to account for 511-KeV photons, but on trying to explain the electron's mass as electromagnetic self-interaction. (See, for example, Lorentz (1904), which gives an expression for the electromagnetic mass in terms of the radius, not calling it the "classical electron radius" yet of course.) XOR'easter (talk) 16:31, 15 May 2020 (UTC)

@XOR'easter: Yes, I agree. The only thing I would add is that if it is true that one century after that much more has been added to our understanding of elementary particles, not much has been added to our understanding of an electric field's energy. That means that if the math of one century ago was telling us that if a positron and an electron kept attracting each other closer than a classical electron radius (from the center of the system, i.e. two electron radii from each other) they would release more energy than the 511 KeV we get – plus any additional energy that you have given to the particles – that math has not changed since. We can come out tomorrow discovering that an electron is a point-like opera singer, but as long as it has an electric charge either we change the math for electric charges or 511 KeV is the energy equivalent of a collision happening at the classical electron radius. --Grufo (talk) 03:42, 16 May 2020 (UTC)

What are you agreeing with? I just said that your portrayal of the history was fundamentally incorrect. And besides that, the material that has been removed and which you say you are "working on" is WP:OR and unsuitable for this project. XOR'easter (talk) 13:51, 16 May 2020 (UTC)

@XOR'easter:

“What are you agreeing with? I just said that your portrayal of the history was fundamentally incorrect.”

I have tried to give you a hand, but you seem blind. The equation for calculating a radius of a mass entirely due to the electrostatic self-energy (the one you are talking about) is:

$r_{\text{e}}=k_{\text{e}}{\frac {e^{2}}{c^{2}m_{\text{e}}}}$       (1)

which is the inversion of:

$m_{\text{e}}=k_{\text{e}}{\frac {e^{2}}{r_{\text{e}}c^{2}}}$       (2)

The equation of calculating half the energy released by two opposite charges that collide at a radius $r$, measured in kilograms instead of joules (the one I am talking about) is:

$M=k_{\text{e}}{\frac {q^{2}}{rc^{2}}}$       (3)

Do (2) and (3) seem different to you? So, the two sentences “the classical electron radius was created with the precise purpose of fitting in that equation and give the electron rest mass as a result” (mine) and “The earliest calculations of a "classical electron radius" of which I am aware were based ... on trying to explain the electron's mass as electromagnetic self-interactionon” (yours) express exactly the same thing: one single identical equation. I had referenced Dirac in “the material that has been removed”, which according to you would be “unsuitable for this project.”: Dirac in 1928 talks about positrons and their relationship with the classical electron radius. Before that date positrons were unknown. Dirac uses the classical electron radius in the equation of a cross section, being any quantum interaction inherently probabilistic. But in a classical interpretation, where no probability is involved, a positron and an electron release their energy at exactly $r_{\text{e}}$. In the electron-positron annihilation according to QED instead, where quanta of known momentum have unknown position, both operators act at the same place. Both QED and classical are models, interpretations. Whatever you adopt, you must be coherent with its math. --Grufo (talk) 15:51, 16 May 2020 (UTC)

I tried to be polite, but this is getting ridiculous. You wrote, All this is obvious, because the classical electron radius was created with the precise purpose of fitting in that equation and give the electron rest mass as a result (because that was what they could see in the experiments: photons of 511 KeV). I pointed out that this is historically inaccurate. Electron-positron annihilation and 511-KeV photons had nothing to do with it; the formula predates even the concept of a photon. You seem to be barreling down a tangent of OR based on a thin thread of dimensional analysis. XOR'easter (talk) 17:43, 16 May 2020 (UTC)

@XOR'easter:

“I tried to be polite”

Let me show you… “The earliest calculations of a "classical electron radius" of which I am aware were based not on electron-positron annihilation or trying to account for 511-KeV photons, but on trying to explain the electron's mass as electromagnetic self-interaction.” -> Me: “I agree” (that's being polite) -> You: “What are you agreeing with? I just said that your portrayal of the history was fundamentally incorrect.”

So no, you didn't even want to. Your being impolite though is completely irrelevant. I will repeat myself: I agree with you, the classical electron radius was created in order to make the electrostatic self-energy entirely accountable for the electron mass. How you get the value of the mass, whether from the drop experiment or from the energy of a photon in a beta decay has never been a point in this discussion. The point is that the value of the mass is obtained experimentally, while the classical electron radius is derived from it (now please feel free to answer again “What are you agreeing with?”).

To derive the classical electron radius it is necessary to satisfy an old equation, much older than the discovery of the electron itself (at least in its original pure-energy form without the mass-energy equivalence). That equation says that the electrostatic self-energy that equals the electron mass, which is 511 KeV, corresponds to a charge contained within a radius $r_{\text{e}}$. That equation also says that the electrostatic free fall of an electron and a positron against each other (from rest) produces 511 KeV of energy (whether via photons or dancing clowns) in a collision happening exactly at $r_{\text{e}}$. And finally that same equation also says also that the merge would produce higher energies if the collision happened at a smaller radius: it is just a consequence of the fact that the charges would acquire more kinetic energy if they approached further.

“You seem to be barreling down a tangent of OR based on a thin thread of dimensional analysis.”

Do you know what dimensional analysis is? How has that anything to do with this?

“this is getting ridiculous”

Again, I agree with you. So I will try to synthesize what I have said so far in order to avoid further misunderstandings:

• An old equation that predates the electron itself allows to calculate the "classical electron radius"

• The same equation says that the electrostatic free fall of an electron and a positron against each other (from rest) produces 511 KeV of energy exactly at $r_{\text{e}}$

• That very same equation also says also that the merge would produce higher energies if the collision happened at a smaller radius

Please refer to my previous answer for further considerations concerning positron discovery and quantum mechanics. --Grufo (talk) 21:11, 16 May 2020 (UTC)

Ignoring what I said, in order to twist it into an agreement with your claim, is not polite. You seem intent upon bringing in ahistorical irrelevancies that do not belong in this or any other Wikipedia article. XOR'easter (talk) 04:23, 17 May 2020 (UTC)

@XOR'easter: Agreeing with you is not ignoring you (and I didn't agree just in part or with a twist, I agreed, full stop). I would rather say that you are constantly failing to acknowledge that. I also think that you keep project on me what you do (“me” not being polite, “me” ignoring you…). That said, what I am agreeing you with is completely offtopic here, this is not a discussion about what experimental method for measuring the electron mass was historically used first, so I should have ignored you rather than agree with you. But fortunately it is not too late for that. --Grufo (talk) 14:57, 17 May 2020 (UTC)

This is a discussion about whether material you wish to add to this article is, in fact, suitable. You made a historical claim ("because that was what they could see in the experiments: photons of 511 KeV"). I interpreted this as an argument you were making in favor of including your material, and I pointed out that it was historically incorrect. If I failed to be clear about this, I apologize. Nothing you have said since has made a case that the disputed text belongs in the article. In addition to the style issues, we need to know: Who has employed that thought-experiment before? Who has argued for its significance? Why is it any more meaningful or helpful or illuminating than the dozens of other situations that one could cook up where the relevant constants are the speed of light and the electron mass and charge, so the relevant length scale will be the classical electron radius up to a 2 or or a π? XOR'easter (talk) 15:12, 17 May 2020 (UTC)

Electrostatic self-energy of the object

I do not understand where does the quotient 3/5 come from.

We know that V = kQ/r and dq = ρ 4πr^2 dr

We also know that dU = Vdq.

Integrating this last equation, we calculate U. So if we put the values of V and dq in U, we obtain

U = Integral(kQ ρ 4πr dr) = kq ρ 2π r^2

Now, we know ρ = q/[(4πr^3)/3] = 3q/(4πr^3). Putting this in U, we finally obtain

U = k (3/2) q^2/r

So I obtain the same equation as in the article, but instead of the quotient 3/5, I obtain the 3/2 one. Where I'm doing maths wrong (assuming the equation of U appearing in the article is correct)? 47.60.54.205 (talk) 17:02, 12 May 2022 (UTC)

Your first expression for U is incorrect. Do the algebra more carefully. Qwerty123uiop (talk) 18:19, 12 May 2022 (UTC)

They are well explained by Feynman here. The constant depends on the charge distribution assumed. Gah4 (talk) 08:24, 14 May 2022 (UTC)

protons

OK, after accidentally putting this in the article, thinking it was in the talk page ...

I suppose, but the classical electron radius is most interesting as it is bigger than the actual electron. Classical proton radius is smaller than the proton. This is related to using wave explanation when the wavelength is larger than the size, and particle explanation when it isn't. Gah4 (talk) 04:37, 11 January 2023 (UTC)