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\section{Theory}  \newcommand{\tensor}[1]{\stackrel{\leftrightarrow}{#1}}  Gauss's Law is in free space is \[\nabla\cdot\vec{E}=\frac{\rho_{ext}}{\epsilon_0}\]. In a plasma $\epsilon_0$ becomes $\tensor{\epsilon}$ and the relation becomes \[\nabla\cdot(\epsilon\vec{E})=\rho_{ext}\] and noting that $\vec{E}=-\nabla\phi$ and tensor $\kappa=\tensor{\epsilon}\epsilon_0$, Gauss's Law can be rewritten as \[\vec{D}=\epsilon_0\tensor{\kappa}\cdot\vec{E}\] where $\tensor{\kappa}$ is the dielectric tensor.   For a cold plasma in an electric field, there will be no contribution to the to the dielectric tensor due to the ions having low thermal velocity in comparison to the electrons. Then the dielectric tensor has components \[\kappa_{\perp}=\kappa_{xx}=\kappa_{yy}=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\] [2]  \[\kappa_{xy}=\kappa_{yx}=\frac{i\omega_{ce}\omega_{ce}^2}{\omega^2-\omega_{ce}^2}\]  \[\kappa_{\parallel}=1-\frac{\omega_{pe}^2}{\omega^2}\]  It is needed to define a potential for an oscillating point charge in this system. Define $\rho$ from Gauss's law as \[\rho_{ext}=qe^{-i\omega t}\sigma(\vec{r})\]