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.