# How does special relativity help us explain the movement of and forces between two charged particles moving in parallel?

I am learning about special relativity, and I just read about how it helps us to solve some "puzzles" that arise in questions involving charged particles. The first "puzzle" was about how a charged particle travelling parallel to a neutral wire behaves, and I understand this completely. What I do not understand is another "puzzle", which involves two like charges travelling in parallel at the same velocity.

The setup is simple: two positive charges are moving in parallel at identical velocities "v". Observer 1 travels at the same velocity, and so observes the particles to be stationary; their repulsion, therefore, is simply explained by electrostatic repulsion. However, for observer 2, who is stationary, the particles are moving, creating an attractive magnetic force between them; despite this, the charges are still repelled.

I have read 3 different explanations from 3 different textbooks:

1. "...the repulsive electric field is increased (through relativistic length contraction). There is also a magnetic field that was not apparent when the observer was stationary relative to the charges. This is because a moving charge gives rise to a magnetic field. Each charge now appears to be moving within the magnetic field due to the other charge and consequently there is an attraction that the observer describes as magnetic in origin. There is both an increased (electrostatic) repulsion and a new (magnetic) attraction compared with the stationary observer frame. Again, a mathematical analysis shows that the force between the charges is identical for all observer inertial frames of reference."

2. "Examining the details of this situation shows that the two observers will reach consistent results only if time runs differently in the two different frames."

3. "This means that an observer in the laboratory will record the electrical force to be just as strong as before but will also record an attractive magnetic force between the two electrons, so the total force will now be smaller than it was in the electrons’ frame of reference. This means that the total force experienced by the electrons depends on the relative velocity of the frame of reference that is being measured. Lorentz calculated the transformation that makes it possible to easily calculate how this force varies from one reference frame to another using the Lorentz factor, γ."

Only explanation 3 makes sense to me, and it seems to be the only one consistent with other answers I've read here (the best example of which is Can relativity explain the magnetic attraction between two parallel electrons or electron beams comoving in a vacuum? (No wires)). But does it not flatly contradict explanation 1? And what is explanation 2 talking about -- how does time dilation have anything to do with this?

These explanations don't really seem to explain what's going on. For explanation 1, why is the electric field increased due to length contraction? I understand that electric field depends on distance between charges, but the distance between them isn't contracting, as contraction only takes place in the direction of motion, and they're travelling in parallel. For explanation 2, I don't see how time dilation has anything to do with the effect. Explanation 3 makes the most sense to me: the electrons can still be repelled for both observers, but the stationary observer will see them being repelled less because of the attractive magnetic force. But how does the Lorentz factor apply here? Would the reduced force calculated by observer 2 be lower by a factor of γ? And if it's true, then it seems to flatly contradict explanation 1.

Any help is appreciated!

The length contracted E field is somewhat colloquial. Yes, a stationary spherical field is squished into a disk at ultra relativistic speeds, but that's just because

$$E'_{\parallel} = E_{\parallel}$$

(parallel to the velocity), while

$$E'_{\perp} = \gamma E_{\perp}$$

for $$B_{\perp} = 0$$. Meanwhile:

$$B'_{\perp} = -\gamma \frac v {c^2} E_{\perp}$$

(Here the charges are stationary in the unprimed frame).

In the primed frame, there is a Lorentz force $$F'=q(E' + v' \times B')$$ that is still repulsive, but you need to use the correct transformations of force/acceleration to get agreement.

So our task is to solve some puzzles involving magnetic fields. The proper way to do that is to put on rubber gloves, grab the magnetic fields and the attached problems, and haul that stuff into a rubbish bin.

Now we have a clean table. So two charges are exerting a force on each other.

Let's ask our buddy A to put two charges at such distance that force is 1N, and hold them there with two hands for one second. Each hand receives an impulse the magnitude of which is 1Ns.

We also ask our buddy B to observe while moving at speed 0.87 c relative to charges. He says that buddy A did put two charges at such distance that force was 0.5 N, and held them there with two hands for two second. Each hand received an impulse the magnitude of which was 1Ns.

That's how it works. I mean relativity has time dilation, and impulse has time and force as factors, and impulse is an invariant.