# Show that for a ferromagnetic material the field in the gap is given by?

## 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.

rude man
Homework Helper
Gold Member
If H = Ni/2piR, then it's kind of hard to believe that H = Ni/l, isn't it?

Hint: what is Ni, which is magnetomotive force, in terms of H(gap), H(material), l and R?

NI is the magneetic force? The number of turns on the toroid multiplied by the current is the magnetic force?

rude man
Homework Helper
Gold Member
NI is the magneetic force? The number of turns on the toroid multiplied by the current is the magnetic force?
I know it by "magnetomotive" force, mmf. It's the magnetic equivalent of electromotive force in electricity.

In electricity, it's emf = -N*∂φ/∂t = iR = El, i = current, R = resistance, φ = magnetic flux, N = no. of turns, R = l/σA, A = area, σ = conductivity, l = path length, E = electric field.

In magnetics, it's mmf = Ni = φR' = Hl, R' = magnetic reluctance, H = magnetic intensity, R' = l/μA, μ = permeability, B = μH.

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