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According to this Veritasium video, the magnetic field in a wire with a non-zero current is an artifact of special relativity. A moving charge sees a speed difference between the wire and the electrons in it (since the electrons are moving). Due to Lorentz contraction there appears to be a net positive charge and a repelling force arises. The physical speed of electrons supporting a current is very slow, a few micrometers per seconds, but due to the number of electrons and the strength of the electric field the effect is macroscopic.

What if you could mechanically increase this speed? Say you had a charged capacitor, and one of the plates was given a parallel velocity. E.g. two concentric cylinders with a charge imbalance, with one of them attached to a motor. Wouldn't that create a very strong apparent magnetic field?

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The magnetic field is created by a current as in the number of electrons per second. The speed of the electrons is thus irrelevant, only the current value in Amperes is important. Therefore the problem you are describing does not exist. However, if you rotate charged objects, such as by attaching them to a fan, they would indeed create a magnetic field. This would not be nearly as efficient as coiling a wire into multiple loops, because the magnetic field is proportional to the number of loops given the same current.

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    $\begingroup$ Increasing the speed of the electrons will increase the current. If the electrons are moving faster, more will travel through in the same amount of time. $\endgroup$ – Johnathan Gross Sep 18 '17 at 1:38
  • $\begingroup$ @Johnathan Gross: Unless the number of electrons drops. Not sure what your point is beyond this trivial arithmetics. $\endgroup$ – safesphere Sep 18 '17 at 1:44
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What if you could mechanically increase this speed? Say you had a charged capacitor, and one of the plates was given a parallel velocity. E.g. two concentric cylinders with a charge imbalance, with one of them attached to a motor. Wouldn't that create a very strong apparent magnetic field?

Well how large is relativistic change if a capacitor plate changes its speed from 0 to 100 m/s? Think about it.

Here's my answer:

At non-relativistic speeds the magnetic force is a very tiny fraction of the electrostatic force. (It's equally small as the length contraction of objects at that speed)

At relativistic speed 0.87 c the magnetic force is 50% of the electrostatic force. If there is an electrostatic force of 1 N between capacitor plates, then when the plates move at speed 0.87 c, there is a 0.5 N magnetic force between those plates. The plates move together.

If only one plate moves, then there is no magnetic force between the plates, because there is only one electric current. (The Veritasium video may disagree with that, which means that the video is not perfectly correct)

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