This solar flare was recorded on July 19, 2012. The color has been added because otherwise we could look at it -you know what they say about looking at the sun. Dr. Phil Plait tells us what's going on here.

What you’re seeing is the profound impact of magnetism on the material in the Sun. I’ve described this effect before (with lots of juicy details here), but in a nutshell: The gas inside the Sun is so hot it’s ionized, stripped of electrons. When that happens it’s more beholden to magnetism than gravity, and when the magnetic field lines pierce the Sun’s surface they form loops along which the ionized gas (called plasma) flows along them.

The total time represented by the video is 21 hours. The flare, though small against surface of the sun, is many times bigger than the earth. Read more about it at Bad Astronomy. Link

The most relevant force in such cases is Lorentz force. In particular, it is mainly the second half dealing with the magnetic field's effect on charged particles, as typically the electric fields within the plasma are small because the plasma is mostly neutral and a good conductor. The plasma consist of a soup of both positive and negative charges all intermixed, so over any macroscopic distance it looks neutral. In fact, if you attempt to create an electric field by adding charge, the plasma will tend to rearrange to block that out (e.g. if you add a positive charge, it will pull a bunch of electrons closer to it, and at a distance, beyond a few millimeters for the solar corona, there will be no net charge seen). So Coulomb's law isn't typically useful over large distances in that particular kind of plasma, although does still get used over very short distances, e.g. working out how collisions between the particles influence things. And gravity would still be relevant here too, even if the magnetic fields strongly constrain the motion. The Wikipedia article on guiding center talks a bit about what happens when you have both a magnetic field and other forces, although I realize neither article I linked has a particularly simple introduction and I don't know of a better site at the moment.

And the real fun starts with more macroscopic effects that can change the magnetic fields and flows together. E.g. too much current in the plasma gives the Kink instability or too much flow causes the firehose instability, and you get wiggles and structure forming along the flows down the field lines.
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I thought that plasma is quasi-neutral chargewise, so Coloumb's law doesn't really apply in the large scale.
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I don't totally follow. Is Coulomb's Law a description of the forces on the particles here, or is some other mechanism in effect in this process?
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Magnetic fields cause loose charge particles to go in circles perpendicular to the field while not applying any forces along the field. The result it is very difficult for them to cross field lines, but very easy to go along the field lines. Although positive and negative particles will go in circles in the opposite direction, the movement along the field lines works the same for both, where they both easily spread out from just pressure.

So if you have heating in the center of the loop, you can keep pushing plasma out of both ends. Although loops on the surface of the sun also have a lot of structure you can't easily see in such videos, due to some of the plasma around them being too cold to light up as much or to the same colors that are being looked at in such videos, so that is another source of material and flow.
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That's pretty kooky, but why does the solar material go both ways on the loop? Don't all those particles have the same polarity?
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