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I've read that an electron beam consisting of electrons that are pretty densely packed manages to stay intact because any one electron experiences a magnetic force caused by the other electrons which somewhat balances the electric repulsion. Of course, that makes sense.

However, that is only true in the ground frame. When we shift over to the frame of any one electron, all electrons would be at rest, because they're all moving at the same velocity. So ideally in this frame, the only force between the electrons would be electrostatic repulsion. And that would mean there is no magnetic force keeping it together.

I've understood that Lorentz forces are supposed to stay the same regardless of the reference frame. However, this doesn't appear to be true, in this situation, since the electric force appears to have equal magnitude in both the frames, while the magnetic force exists only in the first. Can someone highlight where I've gone wrong?

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You have not gone wrong! You just havn't made the final step.

The frame of the bunch of electrons, in which they are stationary, is moving in the lab or ground frame that we live in. So they do repel each other and move apart due to electrostatic forces, (the forces are not equal - the transformation is quite complicated). However we see this (and any related effects) slowed down because of time dilation.

This argument is usually run backwards: if there is Coulomb's law for stationary charges, then charges in moving frames have the effect apparently reduced due to time dilation and one is forced to invent magnetic force to 'explain' this.

Incidentally this is an important effect in particle accelerators. "Space charge" and how to control it are a big topic.

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I've understood that Lorentz forces are supposed to stay the same regardless of the reference frame.

This is where you went wrong. In Special Relativity, forces, including the Lorentz force, are not the same in different inertial reference frames. Accelerations and forces transform under Lorentz transformations. You can find the force transformation in equations (6a) and (6b) in this Wikipedia article.

This is one the many differences between the Lorentz transformations of Special Relativity and the Galilean transformations of Newtonian physics. Accelerations and forces are unchanged by Galilean transformations, but changed by Lorentz transformations.

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A similar discussion in this physics stack exchange channel may answer your question : Can relativity explain the magnetic attraction between two parallel electrons or electron beams comoving in a vacuum? (No wires)

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