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If I put charged particles in the center of long hollow cylinder shaped electromagnet, in a vacuum, would it accelerate them? Is this how particle accelerators work? If so are there formulas I could use to calculate the acceleration(my grandfather might be able to help me, he's an engineer)

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  • $\begingroup$ a traveling wave tube has a longitudinal static magnetic field generated by a solenoid, the electrons are accelerated by a static electric field. The magnetic field is there to keep them aligned with the axis, see en.wikipedia.org/wiki/Traveling-wave_tube $\endgroup$ – hyportnex May 6 '18 at 19:05
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A magnetic field causes charged particles to accelerate, but it does not change their speed. According to the Lorentz force law,

$$\vec{F}=q\vec{E}+q(\vec{v}\times\vec{B})$$

An electromagnet (under DC current)* does not produce any electric field, only a magnetic field, so the $q\vec{E}$ part is zero. Since $\vec{v}\times\vec{B}$ is always perpendicular to the velocity vector, by definition of the cross product, the magnitude of $\vec{v}$ cannot change. Therefore, a magnetic field can only bend the path of charged particles, not change their speed. If you want them to "accelerate" in the sense of a particle accelerator, you're much better off using an electric field (like the one generated between two charged parallel plates).

*An electromagnet under AC current will do something somewhat more complicated, as the changing magnetic field will generate a changing electric field. But for a cylindrical electromagnet under AC current with the magnetic field pointing down the cylinder, the induced electric field will not point down the cylinder, but rather alternately circle clockwise and counterclockwise around the axis of the cylinder. Therefore, this will still not produce behavior consistent with a particle accelerator.

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