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Let's say I am trying to find out the effect that an iron ring has on the Earth's magnetic field (approximated to a uniform field at infinity). Can I solve it in 4 steps:

  1. Find the magnetic field due to a hole of radius R in a flat sheet of iron

  2. Find the magnetic field due to a hole of radius r (where r - R = thickness of ring)

  3. Find the magnetic field due to a plane sheet of metal

  4. Then the field due to the ring is F1 - F2 - F3

I am thinking this will yield the right answer due to the principle of superposition.

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No that doesn't work. If it did, then it should be independent of the specific properties ascribed to the iron - thus should still work if you replaced the iron by air and were asking "What is the effect of an 'air' ring on the Earth's magnetic field"

The obvious answer is "No effect at all", but if you solved each step of your process (with air replacing iron), then F1, F2, and F3 would all be identically +0.4 Gauss. So you would calculate F0 = F1 - F2 - F3 = -0.4 Gauss, with a magical reversal in sign from the true value of F0 = +0.4 Gauss.

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  • $\begingroup$ 1. The sheet of metal would have the same magnetic properties as the iron ring in this scenario 2. There should be some effect of the ring on the nearby fields, since the field lines would prefer to go through the iron rather than free space, it would suck the nearby ones in. $\endgroup$
    – user1402154
    Dec 9, 2020 at 12:33
  • $\begingroup$ My point is that if you replace 'all' the air in 'all' the scenarios, then you should just get F0=Earth's field, but your equation gives F0=(negative)*Earth's field. $\endgroup$
    – Penguino
    Dec 9, 2020 at 20:52
  • $\begingroup$ that is a good point. So I would have to do F1 - F2 - F3 + 2F(earth) $\endgroup$
    – user1402154
    Dec 10, 2020 at 12:53
  • $\begingroup$ At a guess, you just need F1-F2+Fearth, but I am on a mountain bike at the moment and can't check that out exactly. $\endgroup$
    – Penguino
    Dec 12, 2020 at 9:19

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