Ferromagnetic field near a ring in Earth's magnetic field

Question

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:

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

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

Find the magnetic field due to a plane sheet of metal

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.

Penguino · Answer

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.

Ferromagnetic field near a ring in Earth's magnetic field

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