# Light Momentum Transfer between point charges

Hi,

I am trying to understand how light momentum transfers between point charges. I have a few questions if you could help me find some answers or direct me to some sources.

Situation #1
Electron/Wire - There is an electron at rest light years away from a wire that has AC flowing thru it. Because the electron is so far from the wire any magnetic/electric/other forces acting on the wire would be negligible to the electron at rest. When the radiation from the wire reaches the electron would the light momentum accelerate the electron in the direction the light is traveling in.

Situation #2
Electron/Electron - There is an electron at rest light years away from an electron that is accelerating back and forth with a 60 Hz period . Because the electron is so far from the other electron any magnetic/electric/other forces acting on the moving electron would be negligible to the electron at rest. When the radiation from the moving electron reaches the electron at rest would the light momentum accelerate the electron in the direction the light is traveling in.

Situation #3
Electron/Proton - There is an electron at rest light years away from an Proton that is accelerating back and forth with a 60 Hz period . Because the proton is so far from the electron any magnetic/electric/other forces acting on the proton would be negligible to the electron at rest. When the radiation from the proton reaches the electron at rest would the light momentum accelerate the electron in the direction the light is traveling in.

Stavros Kiri

## Answers and Replies

I think there is a radiation interchange between charged particles. One wave from the absorber (traveling backward in time), called an advanced wave, and one from the radiator (traveling forward in time), called a retarded wave.

... I am trying to understand how light momentum transfers between point charges. I have a few questions if you could help me find some answers or direct me to some sources.

When the radiation from the (x3)... ... ... reaches the electron at rest would the light momentum accelerate the electron in the direction the light is traveling in?
Light does indeed carry (energy &) momentum (see my next comment below) and it interacts with matter, so in general the answer to your 3 questions is "yes". But that energy and momentum is radiated and distributed out spherically, so it weakens for large distances. Light years away it will probably be totally negligible (unless the starting one is huge!), thus in practice the answer is "approximately no". Unless you are talking about a laser, which is a totally different story! (Polarized coherent light, which exhibits amplification along the direction of propagation)*. [But then, of course, your examples are not the laser case.]

* In the laser case (in theory and with the proper energy) you can even have a "Laser Blast" acting even light years away! (with time delay of course)

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Light does indeed carry (energy &) momentum (see my next comment below) and it interacts with matter
However, there are 3 main theories in established Physics that study and explain light:
1. Classical E&M theory (i.e. Maxwell's theory) [EM wave momentum density = Const • (ExB) , depending on the unit system]
2. Quantum theory (photons etc. , e.g. due to Einstein, Planck, De Broglie + later developments) [Energy of a photon (of frequency f [and mass m]) E = mc2 = hf , and momentum p= mc = hf/c ]
3. QED (Quantum ElectroDynamics), which is the more correct theory of light, photons and E&M field, and constitutes the Relativistic Quantum [Field] Theory of the Electromagnetic interaction, as studied in modern Quantum Field Theory (e.g. see some of the works of Richard P. Feynman, who was one of the pioneers in the subject in late 20th century [Nobel for QED (shared with 2 others, in 1965)]).

Then, of course, there is 4. Quantum Optics, which is a combination of 2., 3. focussing on optics. [And we forgot 0. Classical Optics [and waves], which barely explains light correctly (mostly phenomenological) ...]

So take your pick ... and study! ...

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I give a correction to myself here:

... ...
... Unless you are talking about a laser, which is a totally different story! (Polarized coherent light, which exhibits amplification along the direction of propagation) ...

Instead, to be more precise, :

... (Polarized, coherent, single frequency light, which exhibits vertical amplification maintained along the direction of propagation) ...

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I think there is a radiation interchange between charged particles. One wave from the absorber (traveling backward in time), called an advanced wave, and one from the radiator (traveling forward in time), called a retarded wave.
In some cases yes, but in this case there is only one wave, because the other side does not oscillate or accelerate.

So... The answer while not trying to be overly concise is yes. There would be acceleration of the light year away electron in the direction the light was traveling.

So... The answer while not trying to be overly concise is yes. There would be acceleration of the light year away electron in the direction the light was traveling.
Yes, but (normally) practically negligible, especially if you do not idealize the situation by assuming vacuum (which is never the case). Only a laser can reach that far, and it has to be strong to move things so far. You see in real space there is still matter that causes diffraction, as well as diffussion and absorbtion. Imagine that just because of diffraction, a well-focussed 2mm apperture 5mW laser, from here to the moon, will spread out its power and energy over 1.6km diameter!
That means in real space even the laser weakens significantly. Imagine then low power ordinary light in light years distance ...
But in ideal vacuum you are right, it will move slightlly.
Also, isn't that how our eyes work? (by accelerating electrons and charged ions inside, caused by the light of distant stars)* [but we are talking about huge star energies to make it here ...]. *Although that may mainly be due to the transverse oscillation due to the wave.

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Thanks for information. Knowing the basic history of how the models are developed give me a lot to look into.

Stavros, you stated that the momentum is proportional to the cross product of the E and B field and thus perpendicular to both in the classical sense. So the momentum would be a vector in the direction the "light" was traveling. Since there is momentum transfer this would also produce "force" that would repel the the two charges.

Situation #4
Electron/Proton accelerating towards each other. Because they are accelerating this would produce "light" transfer/momentum transfer, but because the light is moving between the two would this created a "repulsive" force due to the "light" momentum transfer.

Stavros Kiri
...
... you stated that the momentum is proportional to the cross product of the E and B field and thus perpendicular to both in the classical sense. So the momentum would be a vector in the direction the "light" was traveling. Since there is momentum transfer this would also produce "force" that would repel the the two charges.

Situation #4
Electron/Proton accelerating towards each other. Because they are accelerating this would produce "light" transfer/momentum transfer, but because the light is moving between the two would this created a "repulsive" force due to the "light" momentum transfer.
I like the fact that you know how to ask questions.
Correct about the momentum vector. For classical em, you can also look into "Poynting Vector" (e.g. google - this exact spelling - 'Poynting' is a name). S = E x H (with H = (1/μ)B), connects with (or gives the) "directional energy flux density" for an em field. S is the Poynting vector and it represents an energy flux vector for em energy. Now the density of the linear momentum of the em field is simply: S/c2 .

When we both said "light" in this discussion, I assume that we mean "general em wave" (which could include visual frequencies).

The momentum vector (as well as the Poynting vector) has always the direction of propagation of the wave (as you very efficiently noticed!). And you are correct about the force, but this is not like a typical electrostatic or magnetic repulsive force. It is repulsive due to the "Radiation Pressure" (or the wave's energy and momentum transfer, as you cleverly noticed), and not due to "Polarity". This is an important tricky point here. Thus this force is repulsive even for opposite type (+,-) charges (as long as there is em wave, e.g. due to accelerations). [That's one reason I said that you know how to ask the right questions.]

Now regarding Situation #4 (Electron/Proton ...), I also agree with what you are asserting. Of course one has to take into account the total force for each particle, and it is important to the situation "how far the particles are from eachother" ...

Important Note: The above are based on classical em theory. But, especially for Elementary Particles (such as electrons), or considered within the study of Particle Physics (Particles and Fields and [Quantum] Field Theory), the more correct theory is QED.
Question: do the above conclussions still hold in QED? I think some of them may not. According to R. P. Feynman, the em intetaction between charged particles (e.g. electron and proton) is described by the so-called "Propagator" e.g. in resulting to photon emissions, and one has to study carefully the proper "Feynman Diagrams". (There are many cases there.)
E.g. If there is only one photon and it escapes the particle system then there will be no radiation pressure (on the particles).

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Since your original goal was (is):
I am trying to understand how light momentum transfers between point charges.

I should add that em radiation energy, momentum and pressure [and their transfers] are more obvious and better understood when we know the existence of photons (even in simple [old] Quantum Theory, or better yet in QED). That's because photons are real [energy or field-] particles with real energy and momentum (E = mc2= hf , and p= mc = hf/c) [always moving with the speed of light c].

But of course these are also well understood within classical em theory, via the well-defined notion of em waves, with the help of wave mechanics. Finally, with quantum mechanics, the two notions (of particle and wave) are unified. [And with Quantum field theory, all 3 notions (particle, wave, field) are unified {+with energy (=tot 4) in relativistic theories ...etc.}.]

Could you take time derivative of the "light" momentum vector to get the repulsive force. Also I don't think most waves exert momentum transfer. I'm thinking of a compression wave or an ocean wave because the average position of an object in the wave remains in the same location... If they transfered momentum then the object wouldn't have a constant average locatation. Correct me if I'm not correct about "matter" waves.

Could you take time derivative of the "light" momentum vector to get the repulsive force.
I don't think so (and I havn't seen). Newton's 2nd law does not hold for the em field. But even if it were true you would get a "force density". For the force you would have to integrate over volume (S/c2 is the momentum [of the field] per unit volume). But no. You can't apply Newton's 2nd law here.
Radiation pressure is legitimate (force transferred [on a target] per unit area):