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**The laws of Coulomb versus Ampere and the electromagnetic Machian "paradox"**

Here's an apparent paradox that has been tunneling about in my little mind lately. Maybe someone out there can help me with it.

Imagine if you will, a vast empty region of space devoid of any visible distant stars. In this "laboratory" there exists a very long, very thin rigid rod. The rod is electrically charged with a uniform charge density (dQ/dx) along its length.

An observer next to the rod having no relative motion to the rod measures an electric field from the rod due to the electric charges along the rod. He looks for and finds no magnetic field from the rod which he attributes to the fact that there is no relative motion to the charge along the rod and therefor no electric current density so no induced magnetic field. (Ampere's law of current or Maxwell's 4th eq.)

However when the rod moves along side of the observer (at velocity dx/dt), he does measure a magnetic field since now the charges along the rod are moving relative to him and creating an electric current and consequently a magnetic field. His measurements of the field agree with all known parameters such as the charge density/distribution of the rod, its relative motion to him and the magnetic free space permeability constant. If the observer accelerates to the same velocity as the rod, he sees the magnetic field disappear although the electric field remains. All is well in the observers mind.

Now imagine a second identical rod parallel to the first. It has the same charge density/distribution as the first rod. When both rods move with the same velocity relative to the "static" observer, he measures a magnetic field roughly twice as large as the field from one moving rod which he attributes to the fact that there is twice the current density with two rods and so twice the magnetic field.

The observer realizes that there should be a repulsive force between the two rods created by Coulomb's law relating the force to the square of the ratio of the total charge of the rods and the distance between them. He also realizes that there should be an attractive force between the two rods created by the total magnetic field which is described by Ampere's law relating force to the square of the currents (created by the moving charges: i = dQ/dt = (dQ/dx)(dx/dt)) to the distance between the rods.

The observer calculates that the two forces should be in equilibrium and the rods should remain at a fixed distance from each other when the rods are moving at some relative velocity to him.

The "paradox": But if the observer accelerates to the same velocity as the two rods (dx/dt = 0), he should measure no magnetic field from the rods. Will the two rods now separate since there is no attractive force between them? If so, this seems to violate physics' requirement for an unabsolute or non-fixed reference frame. If the rods don't separate, then why?

Question: If the universe is electrically neutral, is there relativity of charge in motion as there is with mass/energy in motion? By which I mean, would an observer moving along with a charged particle see a relative magnetic field of the universe as he sees relative motion of the universe's mass/energy with respect to him?