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

[blueyellow]

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]

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?

 

[blueyellow]

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

 

[rude man]

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.

 

[paranormal]

Your solution is correct. The field in the gap of a ferromagnetic material is given by B=mu0 N I/l, where mu0 is the permeability of free space, N is the number of loops, I is the current, and l is the length of the gap. This is because ferromagnetic materials have a high permeability, which increases the strength of the magnetic field in the gap. Additionally, the overwound coil creates a strong magnetic field, resulting in a higher value for B. Your explanation and calculation are both accurate and demonstrate a strong understanding of the concept. Keep up the good work!