I Relativity Paradox w/ Charged Spheres

[Xynon]

There is a similar thought experiment I imagined to help me begin to understand the Bell's spaceship paradox:

Consider two positively charged spheres, placed side-by-side inside a frame S' with a string stretched to the point where it balances the repulsive electrostatic force between them. For sake of simplicity, assume that the frame S' and the spheres are perfectly rigid and frictionless. One sphere is directly mounted to the frame S' and the other sphere is tied to the frame S' with a very thin thread same as in the Bell's paradox.

The frame S' starts accelerating relative to the frame S.

According to the frame S', everything is stational and the rope would stay intact. But according to frame S, there would be an attractive magnetic force between the spheres which would break the rope.

 

[mfb]

The image doesn't work.

The force between two accelerated charged objects is a bit more complex than just adding a magnetic term. And the "nice" magnetic term is only present if the objects are at the same coordinate along the acceleration axis - but in that dimension you don't have any length contraction.

 

[Xynon]

The image doesn't work.

The force between two accelerated charged objects is a bit more complex than just adding a magnetic term. And the "nice" magnetic term is only present if the objects are at the same coordinate along the acceleration axis - but in that dimension you don't have any length contraction.

What else comes into play between two accelrated charges? Delayed charge distances/potentials? or EM wave momentum?

I will try to fix the image. In the image, the two charges are at the same coordinate of the acceleration axis (with a distance between them on the perpendicular axis). So wouldn't the magnetic force break the thread? And why no contraction there? We find many examples of charges moving together to have magnetic forces between them. No contraction no magnetic force right?

 

[Xynon]

 

[MikeLizzi]

View attachment 204004

Do you really need to have acceleration? Seems to me you can make your argument by proposing that S' is moving at constant velocity with respect to S.

Propose 2 charged bodies at rest and no magnetic fields with respect to S'.
Then, with respect to S, the charged bodies are moving so there are magnetic fields but there is no apparent change to the magnitude of the electric field to balance the forces.

 

[mfb]

And why no contraction there?

There is no length contraction orthogonal to the direction of motion.

In the frame of an external observer, the repulsion between the charges reduces based on the magnetic field, but the spring gets weaker as well (for the same reason!).

There is no Bell-like paradox here because everything stays simultaneous in all frames as the setup is orthogonal to the line of motion and acceleration.

 

[Nugatory]

There is a similar thought experiment I imagined to help me begin to understand the Bell's spaceship paradox

This thought experiment is about the behavior of electrically charged objects undergoing proper acceleration, so is pretty much unrelated to Bell's spaceship paradox (which is a cleverly disguised exercise in relativity of simultaneity). I have taken the liberty of changing the title accordingly.