# Magnetic Fields in Matter

In dielectric media, dipoles get aligned by an external electric field - a polarization density $$\mathbf{P}\left(\mathbf{r}\right)=\varepsilon_{0}\chi\left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)$$ builds up. In magnetic media, the same happens with the magnetic field and magnetic dipoles. Here, however, the effect is called magnetization $$\mathbf{M}\left(\mathbf{r}\right)=\chi_{m}\left(\mathbf{r}\right)\mathbf{H}\left(\mathbf{r}\right)$$ with the "magnetic suszeptibility" $$\chi_{m}\left(\mathbf{r}\right)$$. Then, the magnetic induction is given by $\begin{eqnarray*}\mathbf{B}\left(\mathbf{r}\right)&=&\mu_{0}\left(\mathbf{H}\left(\mathbf{r}\right)+\mathbf{M}\left(\mathbf{r}\right)\right)\\&=&\mu_{0}\left(\mathbf{H}\left(\mathbf{r}\right)+\chi_{m}\left(\mathbf{r}\right)\mathbf{H}\left(\mathbf{r}\right)\right)\\&\equiv&\mu_{0}\color{red}{\mu\left(\mathbf{r}\right)}\mathbf{H}\left(\mathbf{r}\right)\end{eqnarray*}$with the introduced relative permeability $$\mu\left(\mathbf{r}\right)$$. If $$\mu\left(\mathbf{r}\right)\neq1$$ and so $$\chi_{m}\left(\mathbf{r}\right)$$ does not vanish, we speak of a magnetic medium. In the given problems we will use Maxwell's equations in the magnetostatic approximation to calculate the magnetic field(s) with present magnetic media.