# Magnetic scalar potential for a toroidal ferromagnetic core

1. Oct 3, 2012

### Fabio Riva

1. The problem statement, all variables and given/known data

Given a toroidal core, with known μr, minor radius R1, major radius R2, height h (the section is not a circle, but a rectangle (R2-R1)×h), placed in a magnetic field B0 with cylindrical axisymmetries (B0r=0, B0θ=0, B0z=B0), find the magnetic field resulting by the superposition of the initial magnetic field and the magnetic field generated by the magnetization of the core.
The toroidal core have the same axysimmetry than the field.

2. Relevant equations

To resolve this problem, I use the magnetic scalar potential:
H=-∇ψ → Δψ=∇*M → ρm=-∇*M
So:
ψ=-1/(4π)*∫∇*M/r dV=-1/(4π)∫n*M/r dσ
where the first integral is taken on the volume and the second one on the surface, n is the vector normal to the surface and M is the magnetization of the toroidal core, ∇ is a vector.

3. The attempt at a solution

The problem is divided in two parts: the first is to compute the scalar potential ψ, the second one is to compute the induced field H, and using the superposition with B0 compute the resultant H0.
So, in the computation of ψ, I supposed that M is a vector parallel to z, with norm M.
To compute ψ I have to compute the distance between two point, one (for example P) in the space (outside the core or inside), with parameters (ρ, θ, z) and the other one, inside the core (for exampe Q) with parameters (ρ', θ', z')

The distance PQ is:
rPQ=sqrt((z-z')^2+ρ^2+ρ'^2-2ρρ'cos(θ'-θ))

So, the potential is given by:

ψ(ρ,z)=M/(4π)∫dρ'∫ρ'dθ'/rPQ
with 3 different spaces:

-for ρ>R2 or z>h/2
we have the integral over ρ' taken between R1 and R2, θ' between 0 and2π

-for R2>ρ>R1
the integral over ρ' taken between R1 and ρ, θ' between 0 and2π

-for ρ<R1
whe have ψ=0

Assuming that all these calculations are right, I have a big problem to compute the integral. In fact, knowing the potential ψ, is easy to derive it and find the induced H and after compute the global B, but to derive it I need an analytical solution of the integral. Any idea?