*4.3. Influence of Thin Shell Eddy Current on the Conductor*

The eddy current **J** in the thin shell generates a magnetic potential vector on the conductor:

$$\mathbf{A}\_{\rm EC}(P) = \frac{\mu\_0}{4\pi} \int\_{\Omega} \frac{\mathbf{J}\_{\rm d}}{\mathbf{r}} d\Omega,\tag{23}$$

where r is the distance between the integration point on Ω and the point *P*.

Using (12), Equation (23) can be rewritten as:

$$\mathbf{A}\_{\rm EC}(P) = \frac{\mu\_0}{4\pi} \int\_{\Gamma} \frac{\mathbf{n} \times \mathbf{grad}\Delta\phi}{\mathbf{r}} d\Gamma. \tag{24}$$

Combining (17), (22), (23), Equation (18) is rewritten as:

$$\begin{split} \mathbf{E}\_{\text{ext}}(P) &= \frac{\mathbf{J}(p)}{\sigma} + j\omega \sum\_{k=1}^{\text{m}} \frac{\mu\_{0}}{4\pi} \int\_{\Omega c\_{k}} \frac{\mathbf{J}\_{\text{c}}}{\mathbf{r}} d\Omega + j\omega \frac{\mu\_{0}\overline{\mathbf{C}}}{4\pi} \int\_{\Gamma} \frac{\mathbf{M}\_{\text{r}} \times \mathbf{r}}{\mathbf{r}^{3}} d\Gamma \\ &+ j\omega \frac{\mu\_{0}}{4\pi} \int\_{\Gamma} \frac{\mathbf{n} \times \mathbf{grad}\Delta\phi}{\mathbf{r}} d\Gamma. \end{split} \tag{25}$$
