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Breaking Magnetic Field Lines

When people talk about sunspots and the solar cycle and stuff like that, they often mention magnetic field lines poking through the surface of the sun. I'm pretty sure I get why the magnetic field lines would be twisted up by the non-uniform rotation of the sun, and why they would poke out of the surface.

What I totally *do not* get, is how the heck a magnetic field line *breaks*. This seems to me that it would mean that magnetic field lines are physical things... or rather that they are "made of something", like some kind of material.

I just don't have any sort of idea of how a magnetic field line could be broken based on my understanding of what a field line is. Is this something that I will learn about in E&M courses?

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