Two paralel streams of electrons.

[alpha358]

Consider two parallel streams of electrons in vacuum. Each stream moves with constant velocity and carries a current. According to classical electrodynamics parallel and same direction currents attract each other.
The problem is that I can't see how relativistic length contraction can cause attraction in this situation.

 

[jartsa]

Does classical electrodynamics perhaps say the attractive force becomes equal to the repulsive force when the electrons are moving at the speed of light?


Because in relativity:

Transformed repulsive force = repulsive force / gamma.

Transformed repulsive force approaches zero, when speed of electrons approaches speed of light.

 

[Nugatory]

Consider two parallel streams of electrons. Each stream carries a current.

Ate you asking about two parallel electron beams in a vacuum, or streams of electrons flowing through a current-carrying wire or other conductor?

 

[Dale]

Consider two parallel streams of electrons. Each stream carries a current. According to classical electrodynamics parallel and same direction currents attract each other.
The problem is that I can't see how relativistic length contraction can cause attraction in this situation.

In this circumstance the Lorentz force will never be attractive. As v increases it will become less repulsive, but never attractive. I encourage you to work it out for yourself to confirm.

To understand length contraction's role in reducing the attraction consider the following. Let's say that the spacing between electron's is constant in our frame so that the charge density is constant and I is proportional to v. As v increases the distance between electrons in the electron's frame increases. This is required so that it will length contract down to the correct distance in our frame. That effect causes the acceleration in the electron's rest frame to reduce. Then, that acceleration is further reduced in our frame due to time dilation.

 

[alpha358]

Does classical electrodynamics perhaps say the attractive force becomes equal to the repulsive force when the electrons are moving at the speed of light?

Thanks now I see. I have overlooked the fact that observer at rest will see dominating Coulomb repulsive force and magnetic attractive force increasing with electrons velocity. Sum of these forces is never attractive in this case.