Issue with radiated field, conservation of energy and Poynting vector

[USeptim]

Hello,

I have found an issue in a simple classical electrodynamics problem that I have not been able to explain, so I’m writing this post hopping to find some answer to it.

The problem is this: we have two charged particles with the same charge but different sign, one is massive and I will call it ‘A’ (let’s assume it has infinite mass). The other, ‘B’, has a small mass . The massive particle ‘A’ is in the lab’s frame while the small one ‘B’ is moving towards ‘A’ with a speed v coming from far away and following a trajectory whose minimum distance to ‘A’, with no acceleration, would be d. ‘B’ moves fast enough so that it will not be bundled to ‘A’.

Focusing on particles dynamics, it’s clear that ‘B’ will remain in the same inertial frame all the time since its mass is infinite, so ‘B’ will create a pure Coulomb field that will affect ‘A’s motion. As long as ‘A’ approaches ‘B’, its kinetic energy will decrease being transferred to potential energy. On the other hand, when ‘A’ exceeds ‘B’ and starts to go away. ‘A’ will regain its kinetic energy. Since the Coulomb field is conservative, the potential only depends on distance so ‘A’ will regain all its former kinetic energy as it gets far away from 'B'.

The issue comes now: during the process ‘A’ has been accelerating and therefore radiating power but at the end of the experiment, both ‘A’ and ‘B’ keep its initial kinetic energy so. Where is this radiated energy from?

The Poynting theorem -∂_tu = ∇·S + J·E gives the electromagnetic energy flux plus a term J·E with the work transferred between field and charges.

http://en.wikipedia.org/wiki/Poynting's_theorem

It looks like the J·E term does not take into account the radiation caused by acceleration, in fact it’s the same no matter the ratio between the particle’s charge and mass.

I have read about a “reaction force” to compensate the effect of the radiated field generated from the acceleration but I cannot see as it’s linked to Lorentz Force and Maxwell equations. Any opinions about the validity of this concept?

NOTE: as an axiom (correct me if I’m wrong), the particle’s own field does not affect its motion.

PD: Quantum electrodynimics deals with this problem in the bremsstrahlung process but I’m looking for a solution in the classical model since this issue does not look related neither with quantization of energy nor with small wavelengths.Sergio

 

[soarce]

Please revise your inference because you start with

Hello,
... one is massive and I will call it ‘A’ (let’s assume it has infinite mass). The other, ‘B’, has a small mass.

and afterwards you say

Focusing on particles dynamics, it’s clear that ‘B’ will remain in the same inertial frame all the time since its mass is infinite

.

 

[USeptim]

Hi soarce, you are right, in the third and fourth paragraphs all 'A's and 'B' should be switched, so these paragraphs would be:

Focusing on particles dynamics, it’s clear that ‘A’ will remain in the same inertial frame all the time since its mass is infinite, so ‘A’ will create a pure Coulomb field that will affect ‘B’s motion. As long as ‘B’ approaches ‘A’, its kinetic energy will decrease being transferred to potential energy. On the other hand, when ‘B’ exceeds ‘A’ and starts to go away, ‘B’ will regain its kinetic energy. Since the Coulomb field is conservative, the potential only depends on distance so ‘B’ will regain all its former kinetic energy as it gets far away from 'A'.

The issue comes now: during the process ‘B’ has been accelerating and therefore radiating power but at the end of the experiment, both ‘A’ and ‘B’ keep its initial kinetic energy so. Where is this radiated energy from?​

Next time I will check the post twice... or more...
Sergio

 

[soarce]

Hello,
The issue comes now: during the process ‘A’ has been accelerating and therefore radiating power but at the end of the experiment, both ‘A’ and ‘B’ keep its initial kinetic energy so. Where is this radiated energy from?

Basically you have a time dependent dipole formed by charge A and B which emits electromagnetic radiation. The energy of the emitted radiation comes from the kinetic energy of B and in the end, after B reaches the point of zero velocity, it will have less potential energy than it has had at the stating point (t=0). The energy conservation rule must be applied to the system formed by charges and electromagnetic field.

Related to this kind of phenomena see also the Feynman disk paradox discussed here
https://www.physicsforums.com/threads/feynmans-paradox.399914/

 

[USeptim]

Hi soarce,

The system must keep its total energy but that’s just my concern because I don’t get that result from the classical equations.

The force done by B over A is irrelevant for energy conservation because A has infinite mass.

The force done by A over B is a pure Coulomb force which only depends on B’s relative position. According this force B's kinetic energy at t=∞ will be the same as B initial energy, however, there is no doubt than B's has been radiating energy and this radiation “flees away”, or at least this is how I see things.

The problem seems simple but I got a paradox.

About the Feynman paradox, I have found a good explanation here:

http://maxwellsociety.net/PhysicsCorner/Miscellaneous Topics/FeynmanParadox.html

The paradox comes because of the coil do has angular momentum, not only in the electrons but also from the magnetic field inside the coil. Basically when the current switches off, the angular momentum goes to the disc and accelerates it.

It’s a bit more complicated than my case, however I can’t see how this paradox can answer my doubts.

Sergio

 

[Dale]

NOTE: as an axiom (correct me if I’m wrong), the particle’s own field does not affect its motion.

This is the key problem. If you consider a continuous charge distribution then the particle's own field does in fact affect its motion. But as you take the limit as that charge distribution goes to a classical point charge then some things become undefined and paradoxes arise. So classical point particles are inherently paradoxical, which is one reason that we believe they don't exist.

 

[HomogenousCow]

This is the key problem. If you consider a continuous charge distribution then the particle's own field does in fact affect its motion. But as you take the limit as that charge distribution goes to a classical point charge then some things become undefined and paradoxes arise. So classical point particles are inherently paradoxical, which is one reason that we believe they don't exist.

I thought the issue was solved recently in that book on particles and EM.

 

[Dale]

What book?

 

[soarce]

Hi soarce,
The system must keep its total energy but that’s just my concern because I don’t get that result from the classical equations.

I understand now the issue, one can't formally account for the energy loss in the scenario you proposed.

This is the key problem.

Taking A as finite mass body wouldn't remove the issue? In this case the electromagnetic field produced by B may act on A and in turn the electromagnetic field of A act on B allowing for a kinetic energy change.

 

[Dale]

Taking A as finite mass body wouldn't remove the issue? In this case the electromagnetic field produced by B may act on A and in turn the electromagnetic field of A act on B allowing for a kinetic energy change.

I would have to go through the math to know for sure, but my intuition is that it would not. I believe that the source of the problem is the "radiation reaction". Here is a page that came to my mind when I read this thread:

http://www.phy.duke.edu/~rgb/Class/phy319/phy319/node150.html

 

[USeptim]

This is the key problem. If you consider a continuous charge distribution then the particle's own field does in fact affect its motion. But as you take the limit as that charge distribution goes to a classical point charge then some things become undefined and paradoxes arise. So classical point particles are inherently paradoxical, which is one reason that we believe they don't exist.

Hi DaleSpam,In fact I don't like using point particles since a discrete charge concentrated on a point would lead to an electrostatic field with infinite energy. I prefer thinking in a sphere with all its charge in the Surface. With this model, each charged point would not exert a Coulomb field on the others. Anyway, in order to keep this spherical shape, the Lorentz force caused by external fields should be the same over every point at the sphere and this is an unreal assumption.Sergio

 

[USeptim]

I understand now the issue, one can't formally account for the energy loss in the scenario you proposed.Taking A as finite mass body wouldn't remove the issue? In this case the electromagnetic field produced by B may act on A and in turn the electromagnetic field of A act on B allowing for a kinetic energy change.

Hi soarce,I don't know if turning A into a finite mass would remove the issue. The fact is that it's quite harder to get de dynamics of the particles and the radiated field at the infinite when both A and B have finite mass so I have focused on the more simple case.

A's "infinite mass" could crudely represent a proton against an electron, of course the proton would accelerate but it will suffer an acceleration much smaller than the electron since it's mass is aproximatelly 2000 times bigger.

Sergio

 

[USeptim]

I would have to go through the math to know for sure, but my intuition is that it would not. I believe that the source of the problem is the "radiation reaction". Here is a page that came to my mind when I read this thread:

http://www.phy.duke.edu/~rgb/Class/phy319/phy319/node150.html

I have read some chapters of this very interesting course but not this, I will read it as soon as I can :).

 

[USeptim]

Hi,

I have read the page about the reaction force and though about it. The reaction force let's to keep energy and momentum conservation when there are radiated fields. The price is that you cannot compensate the radiated energy at any instant and therefore you have to integrate over time. This is needed because otherwise, in the particle's innertial frame no force could compensate the radiated energy as F·0 = 0.

In fact this is a low price to pay because higher theories as QED recognice the existence of virtual states that violate temporally conservation laws. The advanced solutions doesn't look very problematic since the time parameter used in the cutoffs is about 10^-24s.

But can this reaction force be connected with Maxwell equations? Becase it looks like a mean to make them fit with conservation laws...
Sergio

 

[Jano L.]

Hi,

I have read the page about the reaction force and though about it. The reaction force let's to keep energy and momentum conservation when there are radiated fields. The price is that you cannot compensate the radiated energy at any instant and therefore you have to integrate over time. This is needed because otherwise, in the particle's innertial frame no force could compensate the radiated energy as F·0 = 0.

In fact this is a low price to pay because higher theories as QED recognice the existence of virtual states that violate temporally conservation laws. The advanced solutions doesn't look very problematic since the time parameter used in the cutoffs is about 10^-24s.

But can this reaction force be connected with Maxwell equations? Becase it looks like a mean to make them fit with conservation laws...
Sergio

The energy of radiation that leaves the system of accelerated particle B and stationary particle A in the frame of A comes from the (kinetic, internal) energy of the particle B and the energy in the EM field in the surrounding space.

In terms of force, elements of the charged particle B experience EM forces due to all other elements and other charged sources and non-EM forces necessary for keeping the particle together. The EM forces are complicated functions of motion of the elements and the non-EM forces do not even have convincing explicit model. The resultant self-force acting on the particle is only approximately given by the Lorentz-Abraham formula.

There is no exact equation of motion for extended charged particles, so there is no way to explicitly express the law of conservation of energy for them. There is only universal view that whatever all the forces are, in the end energy is conserved locally. It is an idea of desirable but non-existing model.

With point particles, everything is much simpler. Poynting theorem does not apply and there is consistent formulation, as in Fokker's, Tetrode's, Frenkel's or Feynman-Wheeler's theory (in the last one, with the requirement of unphysical boundary conditions - perfect absorber - removed).

 

[USeptim]

Hi Jano L,

In brief, we know that energy and momentum must be conservated but our models are not completely accurated. The best model to keep the energy when there is radiation is the Lorentz-Abraham-Dirac force but it's not a perfect model and higher theories such as QED are needed to achieve the energy conservation.

I think I can consider solved my doubt. Thanks to all of you for answering me.Sergio

 

[Jano L.]

Hi Jano L,

In brief, we know that energy and momentum must be conservated but our models are not completely accurated. The best model to keep the energy when there is radiation is the Lorentz-Abraham-Dirac force but it's not a perfect model and higher theories such as QED are needed to achieve the energy conservation.

I think I can consider solved my doubt. Thanks to all of you for answering me.Sergio

The LAD force is an acceptable way to approximately express the additional force for particles with regular charge distribution (no point concentrations of charge). The problem with consistency of this theory is not in lack of quantum theoretical ideas, but in lack of a relativistic model of charged continuum amenable to analysis. I do not think QED addresses this problem of extended particles, far from solving it.

One known way to bypass these difficulties altogether is to discard the idea of extended charged particle and replace it with point particle as the men in my above post have done. There is local conservation of energy in their kind of theory, no QED needed.

 

[USeptim]

The LAD force is an acceptable way to approximately express the additional force for particles with regular charge distribution (no point concentrations of charge). The problem with consistency of this theory is not in lack of quantum theoretical ideas, but in lack of a relativistic model of charged continuum amenable to analysis. I do not think QED addresses this problem of extended particles, far from solving it.

One known way to bypass these difficulties altogether is to discard the idea of extended charged particle and replace it with point particle as the men in my above post have done. There is local conservation of energy in their kind of theory, no QED needed.

In my problem, a point particle will have the same problems than a extended particle.

 

[Jano L.]

In my problem, a point particle will have the same problems than a extended particle.

You need to study theory of point particles:J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.
http://dx.doi.org/10.1007/BF01331692R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Let-
ters, 8, 3, (1964), p. 185-187.
http://dx.doi.org/10.1016/S0031-9163(64)91989-4

J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct
Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.
http://dx.doi.org/10.1103/RevModPhys.21.425

With this view of the theory (no self-interaction), there are no problems of the kind you indicate.