Confused happiness

Recently I’ve been thinking about the magnetic field for a special case of geometry. Basically, if you have a 3D magnetised Pacman, what happens to the magnetic field inside its mouth?

I thought the field would be downwards, since the “upper mouth” is closer to the top of the magnet than the “bottom mouth”. It made sense to me, since magnets can be modelled as charged plates, and if the top mouth is closer to the top of the magnet, shouldn’t it have a higher magnetic charge than the bottom mouth? If so, then the field would flow downwards.

However, today I ran a simulation on that style of geometry. The results showed that the field went upwards inside the mouth. This really worried me, because I was pretty sure my equations wouldn’t work for this type of magnet, rendering them useless for some cases. I was also confused – is the simulation correct? Surely it is. Do I understand magnets? My logic felt so solid, and I started to doubt myself a little.

But it turns out my equations work for this! I was confused because I didn’t understand why the field was upwards, but incredibly happy because my work is correct! I was very concerned about this type of shape (non-convex polyhedron). I knew it worked for convex, and it seems to work for non-convex too!

Later on in the evening, I was of course still thinking about it and why it works the way it does, so I opened up my textbook and started reading. I looked at the basic equations and considered all the variables and what happens to these equations, then it all slowly clicked in my head. It all makes so much sense. Even though the top mouth is closer to the top of the magnet, the outward-facing normal vector is roughly in the opposite direction as the magnetisation vector, meaning that it’s more like a negative magnetic charge.

Today was a bit of a rollercoaster, going from thinking I understood, to being super concerned when the simulation finished, to confused relief when my equations worked, to content when I understood what was going on.

My work seems really promising, and I’m keen to extend it further soon!

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