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What is the mechanism for plasma compression in the video below?

I provide a still from the video and a link to the actual video on YouTube.

enter image description here

https://www.youtube.com/watch?v=gsjWGz3YE5o

BTW, is there a gradient in the magnetic field, meaning that it is stronger near the axis of the glass tube? Like in the picture below?

enter image description here

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This is due to the internal pressure of the plasma. It is a gas, and wants to escape in all directions. However, in the presence of a magnetic field, the Lorentz force comes into play because the plasma is electrically charged.

Consider a single electron in a field that consists of one "magnetic line". In this case the electron will see a force that causes it to orbit the line. Do you see why?

Now consider a whole bunch of electrons in the same field. They all want to do the same thing, but now you also have the repulsion between them to consider. At some critical density, this will overpower the magnetic force.

That's what's happening here. The two magnets produce a field that consists of many "lines of force" running down their open axis - it's a partial solenoid. In this case the area inside each ring has the field is at a maximum and between the two magnets its somewhat less strong.

So the plasma is flowing along these lines but constantly trying to "bust out" across the axis due to standard gas physics. So they fill the area until the pressure drops below the magnetic force in that area. Outside of the rings the field quickly drops and the plasma spreads back out.

This is the classic magnetic mirror setup BTW. In your image the field is weak, so the particles don't reflect at the ends. Of course, in practice, they didn't even with strong fields, which is why no one works on it any more.

enter image description here

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