Magneto Static: solved equations (introduction)


The equations used for the solving of magnetostatic problems are:

  • Maxwell's equations (for a magnetic system)
  • the constitutive equations of the matter

The computation conditions for a magnetostatic application are the following:

  • the state variables are time independent: d/dt = 0
  • the computation concerns only the B and H fields. The D and E fields are not computed. The equations of the electric fields E and D and of the magnetic fields B, H are decoupled.

Equations and conditions

In the previously defined conditions of computation, the equations are summarized as follows:

B: magnetic flux density (in T)

H: magnetic field strength (in A/m)

J: current density (in A/m2)

μ : permeability (in H/m)

The principal equation for magnetic materials, can be put in form B(H) or H(B) as presented below.

μr : relative permeability

μ0 : vacuum permeability

Br : remanent magnetic flux (permanent magnets)


νr : reluctivity νr =1/μr

ν0 : vacuum reluctivity ν0 =1/μ0

Hc : coercive field (permanent magnets)

The models

To solve these equations, two models are used:

  • the vector model, which uses the magnetic vector potential (written )
  • the scalar model, which uses magnetic scalar potentials (written ϕtot or ϕred )

Model and 2D or 3D application

For 2D applications (solved with the 2D solver), only the vector model is available.

For 2D and 3D applications (solved with the 3D solver), the two models (vector and scalar) are proposed:

  • the vector model for 2D applications
  • the scalar model for 3D applications