*4.1. Influence of the Conductor Current on the Thin Region*

The term **H**<sup>0</sup> in Equation (15) can be computed by using Biot and Savart law:

$$\mathbf{H}\_0(P) = \sum\_{k=1}^m \frac{1}{4\pi} \frac{\mathbf{l}\_k}{\mathbf{S}\_k} \int\_{\Omega \mathbf{c}\_k} \frac{\mathbf{l}\_k \times \mathbf{r}}{\mathbf{r}^3} d\Omega\_\prime \tag{19}$$

where I*<sup>k</sup>* and S*<sup>k</sup>* are respectively the current and the cross section of the conductor *k*; Ωc*<sup>k</sup>* is the volume of conductor *k*; **l***<sup>k</sup>* is the vector unit of current direction in the conductor *k* and **r** is the vector between the integration point on Ωc*<sup>k</sup>* and the point *P* where the field is expressed.

Equation (15) is then rewritten as:

$$\frac{\mathbf{M\_{3}}(P)}{\mu\_{r}-1} = \sum\_{k=1}^{m} \frac{1}{4\pi} \frac{\mathbf{l\_{k}}}{\mathbf{S\_{k}}} \int\_{\Omega \mathbf{c\_{k}}} \frac{\mathbf{l\_{k}} \times \mathbf{r}}{\mathbf{r^{3}}} d\Omega - \mathbf{grad} \frac{\overline{\mathbf{G}}}{4\pi} \int\_{\Gamma} \frac{(\mathbf{M\_{3}} \cdot \mathbf{r})}{\mathbf{r^{3}}} d\Gamma + \frac{1}{4\pi} \int\_{\Gamma} \frac{\mathbf{n \times \mathbf{grad}\Delta\phi} \times \mathbf{r}}{\mathbf{r^{3}}} d\Gamma. \tag{20}$$

## *4.2. Influence of Thin Shell Magnetization on the Conductor*

In order to take into account the influence of the field created by the thin shell magnetization, we have to integrate the magnetic vector potential **A**<sup>M</sup> generated by the magnetized thin shell on the conductor. This magnetic vector potential is expressed as [11,24]:

$$\mathbf{A}\_{\mathbf{M}}(P) = \frac{\mu\_0}{4\pi} \int\_{\Omega} \frac{\mathbf{M} \times \mathbf{r}}{\mathbf{r}^3} d\Omega,\tag{21}$$

where **r** denotes the vector between the integration point and the point *P*.

Using (10), Equation (21) can be rewritten as:

$$\mathbf{A}\_{\mathbf{M}}(P) = \frac{\mu\_0 \overline{\mathbf{G}}}{4\pi} \int\_{\Gamma} \frac{\mathbf{M}\_\mathbf{a} \times \mathbf{r}}{\mathbf{r}^3} d\Gamma \tag{22}$$
