A problem about momentum conservation.

[Enryque]

Imagine two equal charges, one at rest and the other moving uniformly. From Special Relativity we know that the electric field of the moving charge is different respect the one of the charge at rest. So the two forces of the interaction do not verify the law of action-reaction and there is a certain amount of mechanical momentum lost.

Where is the mistake?

 

[Orodruin]

The interaction of two charged particles is not instantaneous, this would violate causality. In order to treat this system properly, you cannot ignore the energy and momentum carried by the electromagnetic field. All interactions are local between the charges and the fields and the charges do not interact directly with each other. If the charges are not moving uniformly (which they will not be if they are accelerated by each other's fields), then you cannot just take the "fields of a moving charge" that you will find in the standard textbook and compute the forces of those. You need to apply the Liénard-Wiechert potentials which will be coupled with the equations of motion for the particles themselves.

Edit: Also note that the A-level tag indicates that you have an understanding of the subject at the level of a graduate student or higher. As your questions suggests that you do not, I have changed the thread level to I.

 

[Enryque]

The interaction of two charged particles is not instantaneous, this would violate causality. In order to treat this system properly, you cannot ignore the energy and momentum carried by the electromagnetic field. All interactions are local between the charges and the fields and the charges do not interact directly with each other. If the charges are not moving uniformly (which they will not be if they are accelerated by each other's fields), then you cannot just take the "fields of a moving charge" that you will find in the standard textbook and compute the forces of those. You need to apply the Liénard-Wiechert potentials which will be coupled with the equations of motion for the particles themselves.

Edit: Also note that the A-level tag indicates that you have an understanding of the subject at the level of a graduate student or higher. As your questions suggests that you do not, I have changed the thread level to I.

Thanks for your soon response and adjust the level.

It seems I agree with you about the locality and causality; however the problem is no clear for me. I suppose that this problem can be addressed clasically, with the idea that the force on a charge equals the product of the charge value by the local electric field .

Imagine two external forces such that maintains the motion state of the two charges : namely one at rest and the other with uniform velocity. If we apply the basic mechanics, the external forces must balance the internal ones if the motion state is preserved; and to a external observer, the sum of external forces must be null if the motion of the center of mass is uniform. Also the field is stationary and there is no wave phenomena. So I do not understand well, why the internal forces are not of equal magnitude and opposites.

 

[Orodruin]

I suppose that this problem can be addressed clasically, with the idea that the force on a charge equals the product of the charge value by the local electric field .

No, you cannot address this classically. Electromagnetism is an inherently relativistic theory. If you take the classical limit, the electric field of the charges is the same and you do not have a problem.

Also the field is stationary and there is no wave phenomena

If one of the particles is moving, the EM field is not stationary.

 

[Dale]

So I do not understand well, why the internal forces are not of equal magnitude and opposites

I don't think that this is correct. In the scenario you describe I think that the internal forces are equal.

Have you actually worked out the math on this?

 

[Enryque]

I don't think that this is correct. In the scenario you describe I think that the internal forces are equal.

Have you actually worked out the math on this?

As I belive, the fields of the two puntual charges are, in modulus,like this (R.K. Wangsness - Electromagnetic Fields)

E_rest=q/4πεr ; E_motion=E_restf(v,θ)

if we put the two charges aligned with the relative velocity θ=0 and we have

E_motion=E_rest(1-v²/c²)

so the force on the charge at rest is less intense than the force on the moving charge.

 

[Dale]

As I belive, the fields of the two puntual charges are, in modulus,like this (R.K. Wangsness - Electromagnetic Fields)

E_rest=q/4πεr ; E_motion=E_restf(v,θ)

if we put the two charges aligned with the relative velocity θ=0 and we have

E_motion=E_rest(1-v²/c²)

so the force on the charge at rest is less intense than the force on the moving charge.

I don't know the assumptions of that equation, so I don't trust it. However, I did the calculation using the Lienard Wiechert potential and got a similar result.

However, it is important to note that the direction of the force changes when the charges pass each other. Thus, even though the force on the charge at rest is always less intense, the direction of the change in mechanical momentum flips as the charges pass each other. So on one side there is mechanical momentum going into the fields from the system and on the other side the flow of momentum between the fields and the charges is reversed. Unfortunately, I don't know how to deal with the singularities to demonstrate it, but Poynting's theorem guarantees it.

 

[jartsa]

so the force on the charge at rest is less intense than the force on the moving charge.

Well that is not very true at all, whatever the direction of the motion is.

A moving electric field is kind of concentrated to an area. It's kind of length-contracted.

So therefore:

The force on the charge at rest is more intense than the force on the moving charge, when the charge at rest is located somewhere on the area with the concentrated electric field.

On some other areas the force on the charge at rest is less intense than the force on the moving charge.