<math> \mathbf{H(r)} = \frac{1}{\mu} \nabla \times \mathbf{A} (\mathbf{r}) </math>
Â
The relationship between the magnetic flux density and magnetic field vectors is rather different inside permeable materials that have a permanent intrinsic magnetization. Examples of such materials are ferromagnetic material that are used as permanent magnets. When a permeable material has a permanent magnetization, the following relationship holds:
Â
<math> \mathbf{B(r)} = {\mu} (\mathbf{H(r)} + \mathbf{M(r)} ) </math>
Â
where <b>M(r)</b> is the magnetization vector. In the SI units system, the magnetic field <b>H</b> and magnetization <b>M</b> both have the same units of A/m. It can be shown that for magnetostatic analysis, the effect of the permanent magnetization can be modeled as an equivalent volume current source:
<math> \mathbf{J_{eq}(r)} = \nabla \times \mathbf{M(r)} </math>
Â
If the magnetization vector is uniform and constant inside the volume, then its curl is zero everywhere inside the volume except on its boundary surface. In that case, the permanent magnetization can be effectively modeled by an equivalent surface current density on the surface of the permanent magnetic object:
Â
<math> \mathbf{J_{s,eq}(r)} = \mathbf{M(r)} \times \hat{\mathbf{n}} </math>
Â
where <math> \hat{\mathbf{n}} </math> is the unit outward normal vector at the surface of the permanent magnet object. Note that the volume of the permanent magnet still acts as a permeable material in the magnetostatic analysis.
== Free-Space Magnetic Field and Vector Potential ==