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Update docs/math/maxwell.md
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Co-authored-by: Helge Gehring <[email protected]>
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DorisReiter and HelgeGehring authored Aug 8, 2023
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### Piecewise constant materials and boundary conditions
It is instructive to condsider the well-known case of an interface $I$ between two dielectric materials, which appear in many devices. We assume an interface between two materials called $1$ with dielectric constant $\varepsilon_1$ and $2$ with dielectric constant $\varepsilon_2$. The surface is defined by the normal vector of the interface $\mathbf{n}_{I}$ and there are no external surface charges or currents. For simplicity, we surpress the index $\left(\mathbf{r},t\right)$ here.
It is instructive to condsider the well-known case of an interface $I$ between two dielectric materials, which appear in many devices. We assume an interface between two materials called $1$ with dielectric constant $\varepsilon_1$ and $2$ with dielectric constant $\varepsilon_2$. The surface is defined by the normal vector of the interface $\mathbf{n}_{I}$ and there are no external surface charges or currents. For simplicity, we surpress the arguments $\left(\mathbf{r},t\right)$ here.

All fields can then be split into the component parallel to the interface (hence perpendicular to the normal vector) and perpendicular to the interface (hence parallel to the normal vector). For example we consider the electric field: Define the normalized field vector $\hat{\mathbf{E}}=\mathbf{E}/E$ we split it into

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