Measurements of Resonance Cones and Cone angle in a Steady State Magnetic Field

UCLA Physics 180E



Resonance Cones were observed in a plasma affected by a steady state magnetic field and their angular distribution was measured over a set of electric radio frequencies from the time varying oscillation of a short antenna. The angles were compared to the theoretical values under the cold electron approximation and found to match within a few degrees. Resonance cones at smaller angles were also observed, as predicted by theory for warm electron temperatures where \(\frac{T_{i}}{T_{e}}\ll1\)[3].


Resonance cones are the response in a plasma when there is an oscillating point charge. In order to solve this one needs to solve Gauss’s law with the dielectric constant of a plasma. This gives an equation for electric field and resonance cones are described by the radial component of the electric field. What is discovered is that the electric field forms a cone with cylindrical geometry described by it’s resonance cone angle [1]. Ray Fischer in his thesis paper showed by plotting cone angle with wave frequency for different densities, resonance cones do not behave like a dispersion relation but rather an intereference pattern [1].The sum of the wave numbers for different frequencies interefere to give a large electric field at the resonance cone angle and no electric field at any other positions.


Gauss’s Law in free space is \[\nabla\cdot\vec{E}=\frac{\rho_{ext}}{\epsilon_0}\]. In a plasma \(\epsilon_0\) becomes \({\stackrel{\leftrightarrow}{\epsilon}}\) and the relation becomes \[\nabla\cdot(\epsilon\vec{E})=\rho_{ext}\] and noting that \(\vec{E}=-\nabla\phi\) and tensor \(\kappa={\stackrel{\leftrightarrow}{\epsilon}}\epsilon_0\), Gauss’s Law can be rewritten as \[\vec{D}=\epsilon_0{\stackrel{\leftrightarrow}{\kappa}}\cdot\vec{E}\] where \({\stackrel{\leftrightarrow}{\kappa}}\) 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 \[\kappa_{\perp}=\kappa_{xx}=\kappa_{yy}=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\] \[\kappa_{xy}=\kappa_{yx}=\frac{i\omega_{ce}\omega_{ce}^2}{\omega^2-\omega_{ce}^2}\] \[\kappa_{zz}=\kappa_{\parallel}=1-\frac{\omega_{pe}^2}{\omega^2}\] It is needed to define a potential for an oscillating point charge in this system. Define \(\rho\) from Gauss’s law as \[\rho_{ext}=qe^{-i\omega t}\sigma(\vec{r})\] 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 \(\phi(r,z)=\frac{q}{4\pi\epsilon_{o}\sqrt{\rho^2+z^2}}\) where now \(\rho\) is referring to radius. 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 \[E_r=-\frac{qe^{i\omega t}}{4\pi\epsilon_{0}\kappa_{\perp}\sqrt{\kappa_{\parallel}}}\left(\frac{\rho}{(\frac{z^2}{\kappa_{\parallel}}+\frac{\rho^2}{\kappa_{\perp}})^{3/2}}\right)\] where \(\kappa_{\perp}\) is perpendicular to the background magnetic field. The resonance cones that are observed are described by the electric field in the radial direction. The resonance cone angle is found when looking for at what point does the electric field becomes infinite.This occurs when the condition \(\frac{z^2}{\kappa_{\parallel}}+\frac{\rho^2}{\kappa_{\perp}}=0\). Define \(\tan^2\theta=\frac{\rho}{z}\) then the condition when the radial electric field becomes infinite is when \(\tan^2\theta=\frac{-\kappa_{\perp}}{\kappa{\parallel}}\) where \(\theta\) is the resonance cone angle. This equation can be simplified further by taking the definintions of \(\kappa_{\perp}\) and \(\kappa_{\parallel}\) above and using the condition that \(\omega<\omega_{ce},\omega_{pe}\) called the lower branch [2], then \[\tan\theta\approx f\sqrt{\frac{1}{f_{pe}^2}+\frac{1}{f_{ce}^2}}\] which can also be written as \[\tan^2\theta\approx-\frac{\kappa_{\perp}}{\kappa_{\parallel}}=\frac{(1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2})}{(1-\frac{\omega_{pe}^2}{\omega^2})}\]