Magnetostatics

Ampère's law,

\[\nabla\times\mathbf{B}\left(\mathbf{r}\right)=\mu_{0}\mathbf{j}\left(\mathbf{r}\right)\]

relates the magnetic induction \(\mathbf{B}\left(\mathbf{r}\right)\) to the current distribution \(\mathbf{j}\left(\mathbf{r}\right)\).

In this section we will learn the basic techniques to calculate the magnetic induction for a given current distribution. We will use integral methods like the law of Biot-Savart or differential methods for certain symmetries to solve Ampère's law directly. If you have some nice problems that would fit here or you found some errors, please do not hesitate to contact us!

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.