Exercises

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