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## Homework Statement

consider a toroidal electromagnet with an iron ring threaded through the turns of wire. The ring is not complete and has a narrow parallel-sided air gap of thickness d. The iron has a constant magnetization of magnitude M in the azimuthal direction. Use Ampere's law in terms of the magnetic field vector H, along with the boundary conditions on B and H at the interface, to show that the magnitude of the field H within the iron at a distance r from the centre of the ring is given by

Hiron=(NI-Md)/2πr[/B]

## Homework Equations

B=μ0(M+H)

## The Attempt at a Solution

So far I have calculated the B field in the closed toroid which would be, using Ampere's integral law: B=μ0ΝΙ/2πr.

Based on that plus the fact that Bair⊥=Biron⊥, we assume that for the air the B is the same. Thus, I equated

B=μ0(M+H)→(B-μ0M)/μ0=H→H=(NI-M)/2πr

This is clearly not the correct answer so I would appreciate it if you could please show me how to reach the correct answer.