Capacitors & Co - Dielectrics

The step from the microscopical to the macroscopical Maxwell equations is as fundamental as it is often underestimated. In these problems we will make ourself familiar with the electrostatic version of these equations. We will assume that the electric field induces polarization charges \(\rho_\mathrm{pol}\) in a material as the sources of a polarization density \(\mathbf{P}\),\[\begin{eqnarray*}  \varepsilon_{0}\nabla\cdot\mathbf{E}\left(\mathbf{r}\right)=\rho\left(\mathbf{r}\right)&=&\rho_{\mathrm{ext}}\left(\mathbf{r}\right)+\rho_{\mathrm{pol}}\left(\mathbf{r}\right)\\&=&\rho_{\mathrm{ext}}\left(\mathbf{r}\right)-\mathrm{div}\mathbf{P}\left(\mathbf{r}\right)\ ,  \end{eqnarray*}\]such that we are able to introduce a new field, the electric displacement field \(\mathbf{D}\):\[\begin{eqnarray*}  \nabla\cdot\left[\varepsilon_{0}\mathbf{E}\left(\mathbf{r}\right)+\mathbf{P}\left(\mathbf{r}\right)\right]&\equiv&\nabla\cdot\mathbf{D}\left(\mathbf{r}\right)\\&=&\rho_{\mathrm{ext}}\left(\mathbf{r}\right) \ . \end{eqnarray*}\]Here, \(\rho_\mathrm{ext}\) corresponds to all charges that do not correspond to polarization charges. Using the boundary conditions we can derive from these equations will help us to determine the electrostatic properties of certain dielectric bodies. A special attention will be made to capacitors as these are devices we will be able to describe thoroughly.

Important quantities to characterize dielectric materials are the (relative) permittivity \(\varepsilon\) and the suszeptibility \(\chi\). Both are introduced for linear materials as follows:\[\begin{eqnarray*}  \mathbf{D}\left(\mathbf{r}\right)&=&\varepsilon_{0}\mathbf{E}\left(\mathbf{r}\right)+\mathbf{P}\left(\mathbf{r}\right)\\&=&\varepsilon_{0}\mathbf{E}\left(\mathbf{r}\right)+\varepsilon_{0}\chi\left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)\\&=&\varepsilon_{0}\varepsilon_{r}\left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)\equiv\varepsilon\left(\mathbf{r}\right)\mathbf{E}\left(\mathbf{r}\right)\ .  \end{eqnarray*}\]