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\[\kappa_{xy}=\kappa_{yx}=\frac{i\omega_{ce}\omega_{ce}^2}{\omega^2-\omega_{ce}^2}\]  \[\kappa_{\parallel}=1-\frac{\omega_{pe}^2}{\omega^2}\]  It is needed to define a potential for an oscillating point charge in this system. Define $\rho$ from Gaus's law as \[\rho=qe^{-i\omega t}\sigma(\vec{r})\]  where $\sigma(\vec{r})$ is the delta function at $\vec{r}$ at zero. Use fourier analysis on Gaus's law and note that $E=-\nabla\phi$ to solve for the potential.It is obtained that $\phi(r,z)=\frac{\frac{q}{4\pe\epsilon_{o}}1}{\sqrt{\rho^2+z^2}}$ $\phi(r,z)=\frac{\frac{q}{4\pi\epsilon_{o}}1}{\sqrt{\rho^2+z^2}}$