Magnetic paradox in relativity?

Question

Let 2 electrons A and B be moving parallel with constant velocity $c/10$ in (near) vacuum without a strong gravity field (where $c$ is speed-of-light).

A and B create an electromagnetic field that is proportional to their speed. But then observer A measures ( approx ) the speed of both A and B , and likewise observer B measures ( approx ) the speed of both A and B.

A and B influence each other because of the electromagnetic field differently then if they stood still.

Assuming I made no mistake there , does this not violate relativity since :

1. The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity).
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.

Where did I go wrong? Do the electrons move each other due to the field and does that make me loose the notion of uniform motion relative to one another so that the law does not apply? Or was i wrong about the dependency with speed upon magnetism and electricity ( and right hand rule ) if one uses a time-dilation correction?

In my question I picked $c/10$ to avoid very strong relativistic effects (as with $c/2$) so that classical electromagnetism is still quite good.

I'm confused here.

How to explain this apparent paradox?

Answer 1

This should be on physics.SE, but here goes:

In the lab frame, each electron moves in the other electron's magnetic field, which creates a magnetic attraction between them. Thus, they will be seen to move away from each other slower than their Coulomb repulsion would dictate. (Note, incidentally, that the "Coulomb replusion" wouldn't be the entire story anyway -- the instantaneous electric field of a moving charge differs from the field of a stationary charge at the same place).

In the (initial) rest frame of the electrons, there is no magnetic force, only the Coulomb repulsion, so the electrons move away from each other faster than in the lab frame.

How to reconcile those two computations? Relativistic time dilation comes to the rescue -- the slow separation we see in the lab frame is exactly the fast rest-frame separation, slowed down by the relativistic gamma factor.

Lesson to take home: The net force on each electron is not invariant between frames of reference -- but the entire electomagnetic theory gives consistent answers about the actual movement of charges when calculations are done in different inertial frames.

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By the way, note that classical electromagnetism is not only "still quite good"; it fits perfectly across all speeds up to $c$, since it happens to be invariant under Lorentz transformations (provided that the fields are transformed accordingly) even though it was formulated before relativity.

Answer 2

Ignore magnetism and consider only electrostatitics. In fact, electromagnetism is a relativistic effect. In fact, consider two parallel wires. From the static protons there seems to be no net charge, but the moving electrons view different proton and electron densities and therefore "feel" electrostatic force.

Answer 3

You need to understand that due to relativistic effect from ground frame the charges will experience a greater electric force than when they are at rest. Then they also experience a magnetic force. Add both, it will be equal to Coulomb's force when charges are at rest, as seen from charge frame.