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Chris Spencer edited Theory.tex
about 10 years ago
Commit id: aa7f4d97ffe5d52bfa0447111d4cf1467c70288d
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index 7aa9d3f..93a1c53 100644
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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
$\phi(r,z)=\frac{\frac{q}{4\pi\epsilon_{o}}1}{\sqrt{\rho^2+z^2}}$ $\phi(r,z)=\frac{{q}{4\pi\epsilon_{o}\sqrt{\rho^2+z^2}}$