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I'm really confused with magnetic fields, After reading Purcell's book it seems that the magnetic field is due to choosing the wrong co-ordinate. Then I read this article which says no magnetic field is also a fundamental field. After a discussion on this post; I come up with three ways to produce a magnetic field that are

  1. Moving charge produces a magnetic field
  2. Magnetic field due to intrinsic magnetic moment of particles.
  3. Changing the electric field produces a magnetic field.

I don't know if there are other ways. Now one more thing that confused me is changing the frames. If there is a field in one frame, it may or may not be in another. Two quantities that are invariant under Lorentz transformation are $\mathbf{E}\cdot \mathbf{B}$ and $E^2-B^2$. So from some, the things are clear through this post.

Now my question is How many ways are there through which a magnetic field can exist? By independent I mean here that if it exists in one frame then there is no frame where it doesn't. Please put all ideas that we know till now.

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