Exercises

1. Show that for fields varying as $\exp[\,{\rm i}\,({\bf k}\cdot{\bf r}-\omega\,t)]$ the equations $\nabla\cdot{\bf E}=\rho_c/\epsilon_0$ 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
   $$S^2-D^{2} = R\,L.$$

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

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

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

   $$R=L=S = 1+\frac{{\mit\Pi}_i^{2}}{{\mit\Omega}_i^{2}}+ \frac{{\mit\Pi}_e^{2}}{{\mit\Omega}_e^{2}},$$

   $$D =0,$$

   $$P = -\frac{{\mit\Pi}_i^{2}}{\omega^2} - \frac{{\mit\Pi}_e^{2}}{\omega^2}.$$

8. Show that

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

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

   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
   $$m_s\,\frac{d{\bf V}_s}{dt} + \nu_s\,m_s\,{\bf V}_s = e_s\,({\bf E}+{\bf V}_s\times {\bf B}).$$
   1. Show that the effect of collisions is equivalent to the substitution
      $$m_s\rightarrow m_s\left(1+\frac{{\rm i}\,\nu_s}{\omega}\right).$$

   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

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

      $$k_i \simeq \frac{1}{2\,c}\sum_s \frac{\nu_s\,{\mit\Pi}_s^{2}}{\omega^2},$$

      respectively.
   3. Show that the dispersion relation for a longitudinal electron plasma oscillations is
      $$\omega \simeq {\mit\Pi}_e-{\rm i}\,\sum_s\frac{\nu_s\,{\mit\Pi}_s^{2}}{2\,{\mit\Pi}_e^{2}}.$$
      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

    $$\frac{\partial n}{\partial t} + \nabla\cdot (n\,{\bf V}) =0,$$

    $$m_e\,n\,\frac{d{\bf V}}{dt} =-e\,n\,{\bf E}-\nabla p,$$

    $$p\,n^{-{\mit\Gamma}} =p_0\,n_0^{-{\mit\Gamma}},$$

    $$\epsilon_0\,\nabla\cdot {\bf E} = - e\,(n-n_0).$$

    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 $\omega$.