# Relativistic Coulomb Force

Discussion in 'Physics & Math' started by tsmid, May 16, 2017.

1. ### tsmidRegistered Senior Member

Messages:
368
According to Special Relativity, the usual Coulomb force law is only valid for stationary charges. In case of moving charges, there is an additional factor (see for instance Sect.5.3.4 in http://www.damtp.cam.ac.uk/user/tong/em/el4.pdf )

f=(1-v^2/c^2) / ( 1-v^2*sin^2(θ)/c^2 )^1.5

where v is the velocity of the charge relative to the stationary test charge, θ the angle between v and the radius vector between the two and c the speed of light.

For v/c<<1, we can use a Taylor expansion of the denominator to yield

f ≈ (1-v^2/c^2) * (1+1.5*v^2*sin^2(θ)/c^2) ≈

≈ (1- v^2/c^2*(1-1.5*sin^2(θ)) )

If we have a random distribution for the direction of v, we have to average over sin^2(θ), which yields a constant factor 1/2. So the relativistic factor for an isotropic distribution of the velocity vector is

f_av ≈ (1- v^2/c^2*(1-0.75) ) = 1 - v^2/4c^2

Now for practically all states of matter, the electron speed is several orders of magnitudes higher than that of the ions. The electrons making up atoms and ions for instance have speeds of about 10^6 m/sec , which would imply

1-f_av,el ≈ 3*10^-6

In contrast, 1-f_av for the ions would be several orders of magnitude smaller, so any mass that is neutral at rest, would become charged by a fraction 1-f_av,el. So for instance for a 100 ml glass of water, this would amount to about 16 Coulombs, and two glasses of water at 1 m distance would repel each other with the incredible force of 2*10^12 Newton. And that would be even negligible compared to the forces with which other objects would repel the water and each other. Basically, the whole universe would fly apart.

Any resolutions for this apparent paradox?

3. ### danshawenValued Senior Member

Messages:
3,950
Unless you are using Gryzinski's free fall model of an atom, the electron cloud sees a stationary nucleus with enough positive charges to render the atom as a whole electrically neutral. The electrons in the cloud are anything but stationary. Besides which, the relativistic consequence of a moving charge, in a metallic conducting wire or in free space is a magnetic field, not an increased Coulomb force inside of an atom. What does the moving electric charge see?

Could you describe the relative motion in a little more detail? What exactly is moving with respect to what?

Any ionization involves creating ions of both polarities. If it did not, any chemical reaction involving ion exchange would, as you say, fly apart.

The universe as a whole IS flying apart, just not with as much Coulomb force as you suggest.

Last edited: May 16, 2017

5. ### tsmidRegistered Senior Member

Messages:
368
Any object that sits in front of you apparently at rest consists actually of an equal number of positive and negative charges that are permanently moving around at the microscopic level. The point is that the negative charges (electrons) are moving much faster, so according to Special Relativity their electric field will be different to that of the positive (slower) charges, and thus the object should appear to be charged correspondingly, which is however not observed.

danshawen likes this.

7. ### exchemistValued Senior Member

Messages:
7,340
That is not right, though. There is nothing that reduces the total perceived amount of charge as a result of relativity. The field may look different from a moving frame of reference (leading to the magnetism associated with a moving charge) but there is no reduction in the total charge experienced.

danshawen likes this.
8. ### danshawenValued Senior Member

Messages:
3,950
That's a great argument you have there for Special Relativity playing a role in our rest frame perceptions of atomic structure, charge, and magnetic fields.

I can counter that one only because of a principle discovered in the very late 20th century called the clock postulate.

Newton's calculus is without a doubt the most powerful way to think about things like position, velocity, and in particular, ACCELERATION. Early in your physics study of uniform circular motion, you will learn that in order to produce uniform circular motion, the (linear) velocity vector must always be perpendicular, or at right angles to, the radial vector that points toward the center of rotation, usually visualized as a string under tension as a force counteracting the tendency of the particle to fly off in a straight line.

But acceleration, even in a circular path has precisely squat (nothing, at all) to do with relativistic time dilation (the clock postulate), it turns out. ONLY INSTANTANEOUS VELOCITIES DO. Let that sin in a few moments. What this basically says is, to hell with Newton's conceptions of absolute positions in space and fancy pants mathematical descriptions of acceleration. ONLY relative instantaneous velocities and their directions have anything at all to do with relativity.

The net velocity of any uniform circular or oscillatory motion in the vicinity of an electron cloud surrounding an atom is assuredly zero, as though the entire atom, for the most part, shared our rest frame on average.

Now how well does your relativistic model work?

That was something else that really needed to be said, because even most scientists never fit those particular pieces of the puzzle together. You're welcome.

Last edited: May 17, 2017
9. ### danshawenValued Senior Member

Messages:
3,950
A quantum physicist of course would have explained that probability densities of the waveform for an electron cloud cannot be subjected to relativistic analysis. This is a cop-out. I just did it, and with analytical attitude that has relativistic and mathematical teeth. It isn't that relativity can't explain such things, only that people are retiscent to apply it because it was never really completed.

And the uncertainty principle is a throw-back to Newton's absolute positioning in space. Forget the idea of computing or observing a position to any accuracy on any scale. Quantum physics doesn't even understand all of the ways energy propagates, the speed at which it does so, the boundary conditions or forces necessary for energy to become bound, or entanglement. Yeah, it's a long list, isn't it? With relativity only half complete as well, what does that say about science? We just have a lot of observations we haven't a clue about what to do with.

Last edited: May 17, 2017
10. ### exchemistValued Senior Member

Messages:
7,340
What? Why do you say that? Read this: https://en.wikipedia.org/wiki/Relativistic_quantum_chemistry

Dirac may have been too sweeping to think the effects negligible but he never argued relativity was for some reason inapplicable to QM, as you seem to be asserting.

danshawen likes this.
11. ### danshawenValued Senior Member

Messages:
3,950
Chemistry of course receives a passing grade, and physical chemistry and the technology that has derived from mountains of spectroscopic and chemical observations and analysis over hundreds of years. Those methods and their success is not in question.

Possibly one key to its success is because Newton failed it so miserably. Einstein's improvements to what Newton did just didn't go far enough. What we are left with are large swaths of physics now divorced of any bindings to physical reality. Chemistry obviously would never suffer from that malady, or if it did, not for very long.

12. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member

Messages:
10,450
Oh great another round of useless tilting at windmills by tsmid.

13. ### exchemistValued Senior Member

Messages:
7,340
Eh?

My point is that I do not believe anybody ever suggested QM was exempt from relativity. They simply developed QM without it and it worked. Later, people added its effects in, and were thereby able to explain more phenomena, for instance the colour of gold.

On the - entirely unrelated - issue you raise about Newton and chemistry, I think you are unfair to Newton. My view on that is that chemistry was the last of the physical sciences to be put on a proper footing because it is messy. Until people were able, very painfully, to work out which materials were elements and which were compounds of those elements, it was impossible to make much headway. It was far harder to crack than, say observing and then calculating the motion of the planets, because everyday substances around us are such an unholy mixture.

Mendeleev made perhaps the greatest contribution of them all.

14. ### danshawenValued Senior Member

Messages:
3,950
Yes indeed. The principle strength of chemistry is a large number of scientists working together to fill in the gaps of a body of knowledge that is too big for one scientist to handle. Distributed load works yet again.

This is why it's foolhardy to depend on someone like a Newton or an Einstein to proceed with developing quantum physics as well. It can't help but go badly. We only get one of those individuals once every 500 years or so, even with the best of luck and a stiff wind at our backs. For the most part, we're dullards.

Look, there's a WHOLE OTHER PERIODIC TABLE THERE ON A LOWER SCALE. When I say that bound energy is bound in a composite manner, I mean that in every sense of the word. There are left handed spins rotating with right handed spins, compounded with complex n dimensional spins, that comprise particles of matter and antimatter. We haven't even scratched the surface.

Working on the problem like any of this is derivable by math alone would be as stupid as trying to accomplish the same thing to build up the Periodic Table of the Elements. What is more stupid would be to leave it to the mathematicians to keep doing this while more basic experimental physics remains to be done. I have no doubt, the latter is more likely to be the way this proceeds.

15. ### tsmidRegistered Senior Member

Messages:
368
According to Relativity (I have given the formula with reference in my opening post) the electrostatic force between two charges depends on their relative velocity. So the sum of the electrostatic forces of the same number of positive and negative charges on a given test charge should not add to zero if they have different speeds.

Last edited: May 17, 2017
danshawen likes this.
16. ### tsmidRegistered Senior Member

Messages:
368
I am speaking of instantaneous velocities (be they circular like in an atom or largely rectilinear like in a fully ionized plasma). The point is that the electrons will always have a much higher velocity than the ions, so according to Relativity the electric field produced by the former should be different to that produced by the latter, hence resulting in a net electrostatic force despite the number of charges in each case being the same.

Last edited: May 17, 2017
danshawen likes this.
17. ### danshawenValued Senior Member

Messages:
3,950
The net velocity of the negative charges is zero.

The net velocity of the positive charges is zero.

0 + 0 = 0.

18. ### danshawenValued Senior Member

Messages:
3,950
So am I. The net INSTANTANEOUS velocity of the negative charges is zero. They travel in circles, like a rotation. They travel very fast in a very confined (Angstroms) space. For any given instant, no net motion in any direction that is not balanced the next instant by one in the opposite direction.

Not being a wise guy here. That's just how uniform circular motion works.

Neglecting forces arising from any residual (frictional) static electric charges, the net charge is still zero.

19. ### tsmidRegistered Senior Member

Messages:
368
The net velocity may average to zero but the net electric field isn't. Check out my opening post again. Since the angle θ appears only as a square function it can not average to zero.

danshawen likes this.
20. ### danshawenValued Senior Member

Messages:
3,950
The net electric current in a conducting wire not connected to a current source is zero even though there may be residual free charge in the electron gas of the conductor, as the theory goes.

All that remains of electric charge mobility is termed "thermal noise", "shot noise", or perhaps noise induced by means of radio or microwaves, with the conductor as an antenna. None of that creates very much of a force, let alone the magnitude of force your opening post about separating the charges in a glass of water would suggest. You can actually rectify some of those things and produce electricity by means of a thermocouple, a "crystal" radio receiver, etc. All good fun. And it all works. It wouldn't, if what you suggest were true.

But you have not demonstrated any net velocity capable of doing anything like separating out a large amount of electric charge from its opposite polarity in atomic structure. Relativity won't be able to do it.

21. ### tsmidRegistered Senior Member

Messages:
368
If the net velocity is zero then obviously the net current is zero as well (in a current carrying wire the net velocity is not zero). But as per the velocity dependent Coulomb force (as given in my opening post) the net electrostatic field should depend on the velocity even if the net velocity is zero (as the contributions from different directions of the velocity vector do not cancel because of the term sin^2(θ). This is what Relativity is implying not I.

danshawen likes this.
22. ### exchemistValued Senior Member

Messages:
7,340
This is interesting. I do not pretend to know much about relativity but I wonder if one problem is your attempt to apply special relativity to a system in which the moving charges are in orbital motion about the stationary one, rather than linear motion.

This looks to me like a situation in which the Ehrenfest paradox may come into play. As I understand it, the perceived charge density (field strength) changes due to Lorentz contraction in the direction of perceived motion. So you have, I think, the same issue as the length contraction in the periphery of a rotating disc, which is the subject of that paradox. I cannot resolve this for you but perhaps someone familiar with this paradox may be able to help.

danshawen likes this.
23. ### danshawenValued Senior Member

Messages:
3,950
A net velocity of zero is "at rest", for all intents and purposes, even though for a free electron, the state of motion "at rest" means that it literally and figuratively ceases to exist (waveform collapses, as per quantum physics). If the waveform for the electron did not collapse did not, then it violate the uncertainty principle.

It is the incompatibility of the uncertainty principle with the rest frame of relativity that confuses some people, like me, for instance. Absolute position is impossible ever to know, for anything other than the Lorentz roadbed under relativity, but evidently, it's okay again for quantum physics.

For certain, there are cracks (big ones) in our understanding of physics on different scales. There was a ready answer to yours. For the most part, ideas like that are easily ignored by people who believe their math informs them of something reality doesn't. Things like dividing by zero in proportional math, I am confident, informs you of exactly nothing.

I like the way you think. Keep up the good work.

Last edited: May 18, 2017