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## Homework Statement

A toroidal shaped magnetic material of radius, a, and cross sectional radius R has a small transverse gap cut into it of length l. The toroid is uniformly overwound with a coil of N loops carrying a current I. Show that for a ferromagnetic material the field in the gap is given by

B~mu0 N I/l

[5 marks]

## The Attempt at a Solution

Assume that the curvature is small, so that locally B, H and M are parallel to each other, uniform across the cross-section and tangential. A circular loop integral of radius R will have the same value of H at every point, so:

the loop integral of H.dl=H 2 pi R=NI

H=NI/2 piR

2 pi R is the circumference, so for a toroid with a gap of length l:

H=NI/l

B=mu0(H+M)

M=0 because the gap is made of empty space

B=mu0 (H+0)

=mu0 NI/l

Have I gone wrong somewhere by saying something that's not true? I'd be grateful if you could help please.