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Chris Spencer edited Theory.tex
about 10 years ago
Commit id: 75d39b022d6fae046d31b734f636cabd57fa7e4e
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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 $\phi(r,z)=\frac{q}{4\pi\epsilon_{o}\sqrt{\rho^2+z^2}}$. The resonance cone phenomena is described by electric fields so take the negative gradient of $phi$ in cylindrical coordinates and the electric field is in the radial direction is \[E_r=-\frac{qe^{i\omega
t}}{4\pi\epsilon_{0}\kappa_{\perp}\sqrt{\kappa_{\parallel}}}\left(\frac{\rho}{(\frac{z^2}{\kappa_{parallel}}+\fraqc{\rho^2}{\kappa_{perp}})^{3/2}}\right)\] t}}{4\pi\epsilon_{0}\kappa_{\perp}\sqrt{\kappa_{\parallel}}}\left(\frac{\rho}{(\frac{z^2}{\kappa_{parallel}}+\frac{\rho^2}{\kappa_{perp}})^{3/2}}\right)\]