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When I imagine a magnetic field produced by a magnet, or the electric field produced by a charge, I've learned that the fields are stationary, however, their value(across space) changes.

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If I placed the magnet at a point $P$($0,0,0$), and then moved the magnet to $P_2$$(1,1,1)$ Why wouldn't it's associated magnetic field move with it?

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Fields permeate infinite space and always have a value at each point. The only sense in which they can move is that those values can change with time. For example, a field can become strong in a place where it was previously weak. So you could talk about, say, the maximum of the field moving, but the field itself isn’t a localized thing that can move. It’s already everywhere!

By the way, there actually is only one electromagnetic field in the universe. If you have, say, two magnets, they just make that one field strong near them. They don’t actually have “their own” magnetic fields, although we often talk as if they do when we calculate things.

Every photon in the universe is a quantum of this one universal electromagnetic field!

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If the magnet is moved, the magnetic field changes to match the new position of the magnet. As the magnetic field changes, it also induces an electric field.

The details of this interaction depend heavily on how exactly the magnet is being moved, and may or may not involve the production of propagating electromagnetic waves (for example, an accelerating magnet will generate electromagnetic waves, while a magnet moving at a constant velocity will not).

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The fields are stationary with respect to a coordinate system fixed to the magnet.

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