How can a point charge exert force on itself?

[maline]

An accelerated charge emits radiation and so must lose energy. This implies that it feels a "reaction force" in the direction opposite the motion. Since EM interactions conserve energy/momentum, it must be possible to describe the reaction force in terms of EM fields acting on the charge. These fields must be generated by the charge itself- we may assume that any external fields are uniform.
Now, we know that the fields at any point can be found by integrating a function of the charges & currents along the past (or future) light-cone of that point- the "retarded time" integral. But our particles world-line is of course timelike, so it never intersects the light-cone of any point on the path!
So how can the particle produce a force on itself?

 

[Vanadium 50]

So how can the particle produce a force on itself?

It doesn't.

 

[maline]

Care to actually answer the question?

 

[BvU]

Question has been answered. What you want to hear is where your reasoning fails, right ?

 

[maline]

Yes, obviously. Pardon me for being annoyed, but no response at all is clearly better that a response that clearly won't help me.

 

[BvU]

Well, perhaps V50 will help us out. I've grown accustomed to the idea that there's no point in wanting to know about the self-energy of a point charge (renormalization and all that), but maybe there's a better story to be told !
At least your slight sarcasm demonstrates you care !

 

[maline]

Just to clarify, I am asking about the classical version of the problem. I know there is an ODE for the reaction force in terms of the derivative of the acceleration, but it seems that consistency requires a formulation in terms of the self-field, hence my question: it seems the self-field at the particle's position cannot depend on the history, (and certainly not the third derivative of the position), because the world-line does not intersect the past light-cone.

 

[vanhees71]

This is a very tough issue, which is not completely solved and in my opinion hardly ever completely solvable, because classical point particles are ill-defined in relativistic field theories. My favorite paper on the topic at the moment is

http://arxiv.org/abs/physics/0508031

 

[tech99]

An accelerated charge emits radiation and so must lose energy. This implies that it feels a "reaction force" in the direction opposite the motion. Since EM interactions conserve energy/momentum, it must be possible to describe the reaction force in terms of EM fields acting on the charge. These fields must be generated by the charge itself- we may assume that any external fields are uniform.
Now, we know that the fields at any point can be found by integrating a function of the charges & currents along the past (or future) light-cone of that point- the "retarded time" integral. But our particles world-line is of course timelike, so it never intersects the light-cone of any point on the path!
So how can the particle produce a force on itself?

Sorry, I am just an amateur scientist, so probably wrong, but my simple perception was that the charge has lines of electric force radiating out in all directions, and all pulling outwards so that the total force is zero. If the charge is suddenly accelerated, the lines are distorted. This is because the new information propagates outward at a finite speed of c. The distortion of the lines is such that they are now pulling back, against the acceleration. Like a ball hitting a net. So we end up doing work against the attraction of the field lines and this energy is lost as radiation. Maybe someone can correct my picture in a simple way?

 

[phyzguy]

I recommend reading The Feynman Lectures on Physics, Vol 2, Chapter 28 for an excellent summary of the classical attempts to deal with these questions.

 

[maline]

Sorry, I am just an amateur scientist, so probably wrong, but my simple perception was that the charge has lines of electric force radiating out in all directions, and all pulling outwards so that the total force is zero. If the charge is suddenly accelerated, the lines are distorted. This is because the new information propagates outward at a finite speed of c. The distortion of the lines is such that they are now pulling back, against the acceleration. Like a ball hitting a net. So we end up doing work against the attraction of the field lines and this energy is lost as radiation. Maybe someone can correct my picture in a simple way?

As you said, the lines are radiating out at speed c. So the particle can't possibly catch up with them!
If the particle has a finite radius, then one side can be slowed down by the field of the other side. The question is what happens when or if we let the radius go to zero...

 

[tech99]

As you said, the lines are radiating out at speed c. So the particle can't possibly catch up with them!
If the particle has a finite radius, then one side can be slowed down by the field of the other side. The question is what happens when or if we let the radius go to zero...

Under static conditions, my picture was that the electric field lines just extend out like radial strings and are not moving outward. When acceleration occurs, some of the strings are bent, so that they are at an angle to the charge and no longer apply a balanced pull on the charge. Maxwell's vision of the lines of force was of elastic strings under tension.

 

[maline]

Of course, electric fields never actually move. They are just vector-valued functions of position & time, that describe the force on a test particle. "Lines of force" is just a descriptive way of saying "curves that are tangent to the field at each point". But as you mentioned, the influence of charges & currents on the field propagates at speed c. Technically this is true even in the the static case: the field around a charge is kq₁q₂/R² "because" the charge was in its position R/c seconds ago, not because it's there now.
As for "elastic strings under tension", I don't see at all what might have been intended by that, so I can't tell you definitively that it's wrong. What I can tell you is that it bears no reseblence to the way fields are normally spoken about & understood, and if you want to study EM you will probably need to drop that idea.

 

[David Lewis]

I don't think the point charge can exert force on itself, and I don't see how it could interact with empty space*, so the only thing left is radiation being emitted by other point charges. If we have a retarded wave (arrives after it leaves the source) then there should also be an advanced wave coming from somewhere else (arriving before it leaves the source).

*There would have to be at least one other electron in the universe to absorb the radiation.

 

[tech99]

I don't think the point charge can exert force on itself, and I don't see how it could interact with empty space*, so the only thing left is radiation being emitted by other point charges. If we have a retarded wave (arrives after it leaves the source) then there should also be an advanced wave coming from somewhere else (arriving before it leaves the source).

*There would have to be at least one other electron in the universe to absorb the radiation.

Under static conditions, my picture was that the electric field lines just extend out like radial strings and are not moving outward. When acceleration occurs, some of the strings are bent, so that they are at an angle to the charge and no longer apply a balanced pull on the charge. Maxwell's vision of the lines of force was of elastic strings under tension.

I have shown a demonstration of a mechanical analogue of radiation. The Slinky represents a conductor, and I have sent a longitudinal wave of compression along it. The string which is hanging represents a static field line from an electron. As the wave on the wire passes, it accelerates the electron, and a transverse ripple is sent out on the static field line, representing radiation. Although this energy leaves the wire at the speed of propagation on the string, the electron still feels a force opposing its acceleration, against which the generator must do work.
https://www.flickr.com/photos/58337586@N08/26137254760/in/dateposted-public/

 

[marcusl]

Just because you constructed a mechanical model of EM fields doesn't make it correct. Field lines are not strings.

 

[tech99]

Just because you constructed a mechanical model of EM fields doesn't make it correct. Field lines are not strings.

It is meant to portray classical physics, such as the views of J J Thompson (who suggested acceleration of a charge as a mechanism of radiation) and Joseph Larmor. The formula which Larmor produced for radiation seems still correct today.

 

[marcusl]

I don't follow. To my knowledge, neither Thomson (no 'p') nor Larmor proposed an elastic string model of EM fields.

 

[David Lewis]

In this thread, the model only needs to be realistic enough to answer the OP's question.

 

[tech99]

I don't follow. To my knowledge, neither Thomson (no 'p') nor Larmor proposed an elastic string model of EM fields.

The basis of my model is the book "Electromagnetic Vibrations, Waves and Radiation", by Bekefi and Barrett of MIT. Lines of force must be elastic because if two opposite charges move apart the string does not break.

 

[marcusl]

The model has the potential to mislead. Elastic bands immediately bring to mind force, but the behavior is all wrong. The force of an elastic increases linearly with distance between the ends, while the Coulomb force falls quadratically with separation. I think it needs to be discussed with care.

 

[tech99]

The model has the potential to mislead. Elastic bands immediately bring to mind force, but the behavior is all wrong. The force of an elastic increases linearly with distance between the ends, while the Coulomb force falls quadratically with separation. I think it needs to be discussed with care.

Agree

 

[Jano L.]

An accelerated charge emits radiation and so must lose energy.

If we assume only retarded fields:

If the accelerated charged body consists of smaller elements held together and interacting with each other, there indeed will be EM energy propagating out away from the body. This energy is accounted for by decrease of EM energy near the body and decrease of kinetic and internal energy of the particles forming the body - the internal forces do work on each other and resist acceleration.

If the accelerated charged body is point particle, it has no parts and it has nothing to interact with. There is no reason to believe then its radiation by itself is associated with EM energy propagating out away. (Poynting's theorem does not hold in point particles, although Maxwell's equations are fine).

 

[vanhees71]

Point particles lead to severe mathematical problems in classical electrodynamics (or any relativistic classical field theory). It simply doesn't make sense. It's always to be defined as a certain limit of extended charged bodies, and one has to take into account the Poincare stresses to hold this extended charged body together. Using a quasi-rigid model, the following paper gives a very nice treatment of the problem:

http://arxiv.org/abs/physics/0508031

 

[tech99]

If we consider the matter of reciprocity between transmitting and receiving antennas, are you saying that the point charge will not move in response to an incoming EM wave? I tend to doubt this.

 

[vanhees71]

Of course it will move, and as long as you neglect the "radiation reaction", there's no problem to solve for the equations of motion of this point charge in the given electromagnetic field (including those of incoming em. waves). This is underlying the design of all particle accelerators, including the LHC at CERN!

 

[Jano L.]

Point particles lead to severe mathematical problems in classical electrodynamics (or any relativistic classical field theory). It simply doesn't make sense. It's always to be defined as a certain limit of extended charged bodies, and one has to take into account the Poincare stresses to hold this extended charged body together. Using a quasi-rigid model, the following paper gives a very nice treatment of the problem:

http://arxiv.org/abs/physics/0508031

The severe mathematical problems are actually due to misuse of the Poynting formulae (quadratic functions of fields). Maxwell's equations as well as the Lorentz force work very well with point particles (Lienard-Wiechert fields of point charge are solutions of Maxwell's equations, and the most used form of the Lorentz force formula assumes particle is point).

Extended charged bodies are different and more complicated, because they have parts and infinite number of degrees of freedom. Limit of their theory does not lead to sane theory of point particles, because the limit does not reflect the fact point particles do not have parts.

 

[Jano L.]

If we consider the matter of reciprocity between transmitting and receiving antennas, are you saying that the point charge will not move in response to an incoming EM wave? I tend to doubt this.

You're right, the particle will respond to external EM field.