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Next: Wave Propagation Through Inhomogeneous Up: Waves in Cold Plasmas Previous: Perpendicular Wave Propagation


  1. Show that for fields varying as $ \exp[\,{\rm i}\,({\bf k}\cdot{\bf r}-\omega\,t)]$ the equations $ \nabla\cdot{\bf E}=\rho/\epsilon_0$ and $ \nabla\cdot{\bf B}=0$ follow from Equations (5.4) and (5.5). This explains why the former equations are not explicitly used in the study of plane waves.

  2. Derive Equations (5.31)-(5.33) from first principles, starting from the equations of motion of individual charged particles.

  3. Prove the identity

    $\displaystyle S^2-D^{\,2} = R\,L.

  4. Derive the dispersion relation (5.44)-(5.47) from Equation (5.42).

  5. Show that the square of $ F$ , defined in Equation (5.48), can be written in the positive definite form

    $\displaystyle F^{\,2} = (R\,L-P\,S)^2\,\sin^4\theta + 4\,P^{\,2}\,D^{\,2}\,\cos^2\theta.

  6. Derive the alternative dispersion relation (5.50) from (5.44).

  7. Show that in the limit $ \omega\rightarrow 0$ ,

    $\displaystyle R=L=S$ $\displaystyle = 1+\frac{{\mit\Pi}_i^{\,2}}{{\mit\Omega}_i^{\,2}}+ \frac{{\mit\Pi}_e^{\,2}}{{\mit\Omega}_e^{\,2}},$    
    $\displaystyle D$ $\displaystyle =0,$    
    $\displaystyle P$ $\displaystyle = -\frac{{\mit\Pi}_i^{\,2}}{\omega^2} - \frac{{\mit\Pi}_e^{\,2}}{\omega^2}.$    

  8. Show that

    $\displaystyle \frac{{\rm i}\,V_{x\,i}}{V_{y\,i}}$ $\displaystyle = \frac{({\rm i}\,E_x/E_y)- ({\mit\Omega}_i/\omega)}{1-({\mit\Omega}_i/\omega)\,({\rm i}\,E_x/E_y)},$    
    $\displaystyle \frac{{\rm i}\,V_{x\,e}}{V_{y\,e}}$ $\displaystyle = \frac{({\rm i}\,E_x/E_y)-({\mit\Omega}_e/\omega)}{1-({\mit\Omega}_e/\omega)\,({\rm i}\,E_x/E_y)}.$    

    Hence, deduce that for a right-hand/left-hand circularly polarized wave the ions and electrons execute circular orbits in the $ x$ -$ y$ plane in the electron/ion cyclotron direction.

  9. The effect of collisions can be included in the dispersion relation for waves in cold magnetized plasmas by adding a drag force $ \nu_s\,m_s\,{\bf V}_s$ to the equation of motion of species $ s$ . Here, $ \nu_s$ is the effective collision frequency for species $ s$ , where $ s$ stands for either $ i$ or $ e$ . Thus, the species $ s$ equation of motion becomes

    $\displaystyle m_s\,\frac{d{\bf V}_s}{dt} + \nu_s\,m_s\,{\bf V}_s = e_s\,({\bf E}+{\bf V}_s\times {\bf B}).

    1. Show that the effect of collisions is equivalent to the substitution

      $\displaystyle m_s\rightarrow m_s\left(1+\frac{{\rm i}\,\nu_s}{\omega}\right).

    2. For high frequency transverse waves, for which $ \nu_s\ll \omega$ , and $ {\mit\Pi}_e, \vert{\mit\Omega}_e\vert\ll \omega$ , show that the real and imaginary parts of the wavenumber are

      $\displaystyle k_r$ $\displaystyle \simeq \frac{\omega}{c}\left(1-\frac{{\mit\Pi}_e^{\,2}}{2\,\omega^2}\right),$    
      $\displaystyle k_i$ $\displaystyle \simeq \frac{1}{2\,c}\sum_s \frac{\nu_s\,{\mit\Pi}_s^{\,2}}{\omega^2},$    

    3. Show that the dispersion relation for a longitudinal electron plasma oscillations is

      $\displaystyle \omega \simeq {\mit\Pi}_e-{\rm i}\,\sum_s\frac{\nu_s\,{\mit\Pi}_s^{\,2}}{2\,{\mit\Pi}_e^{\,2}}.

      Hence, demonstrate that collisions cause the oscillation to decay in time.

  10. A cold, unmagnetized, homogeneous plasma supports oscillations at the plasma frequency, $ \omega={\mit \Pi}_e$ . These oscillations have the same frequency irrespective of the wavevector, $ {\bf k}$ . However, when pressure is included in the analysis, the frequency of the oscillation starts to depend on $ {\bf k}$ . We can investigate this effect by treating the (singly-charged) ions as stationary neutralizing fluid of number density $ n_0$ . The electron fluid equations are written

    $\displaystyle \frac{\partial n}{\partial t} + \nabla\cdot (n\,{\bf V})$ $\displaystyle =0,$    
    $\displaystyle m_e\,n\,\frac{d{\bf V}}{dt}$ $\displaystyle =-e\,n\,{\bf E}-\nabla p,$    
    $\displaystyle p\,n^{\,-{\mit\Gamma}}$ $\displaystyle =p_0\,n_0^{\,-{\mit\Gamma}},$    
    $\displaystyle \epsilon_0\,\nabla\cdot {\bf E}$ $\displaystyle = - e\,(n-n_0).$    

    where $ p_0$ and $ {\mit\Gamma}=5/3$ are constants. Let $ n=n_0+n_1$ , $ p=p_0+p_1$ , $ {\bf V} = {\bf V}_1$ , and $ {\bf E}={\bf E}_1$ , where the subscript 0 denotes an equilibrium quantity, and the subscript $ 1$ denotes a small perturbation. Develop a set of linear equations sufficient to solve for the perturbed variables. Assuming that all perturbed quantities vary in space and time as $ \exp[\,{\rm i}\,({\bf k}\cdot{\bf r}-\omega\,t)]$ , find the dispersion relation linking $ \omega$ and $ {\bf k}$ . Find expressions for the phase-velocity and group-velocity of the wave as functions of $ \omega$ .

next up previous
Next: Wave Propagation Through Inhomogeneous Up: Waves in Cold Plasmas Previous: Perpendicular Wave Propagation
Richard Fitzpatrick 2016-01-23