Exercises

1. Consider a quasi-neutral plasma consisting of electrons of mass $m_e$, charge $-e$, temperature $T_e$, and mean number density, $n_e$, as well as ions of mass $m_i$, charge $Z\,e$, temperature $T_i$, and mean number density $n_i=n_e/Z$.
   1. Generalize the analysis of Section 1.4 to show that the effective plasma frequency of the plasma can be written
      $${\mit\Pi} = \left({\mit\Pi}_e^{2} + {\mit\Pi}_i^{2}\right)^{1/2},$$
      where ${\mit\Pi}_e= (e^2\,n_e/\epsilon_0\,m_e)^{1/2}$ and ${\mit\Pi}_i = (Z^{2}\,e^2\,n_i/\epsilon_0\,m_i)^{1/2}$. Furthermore, demonstrate that the characteristic ratio of ion to electron displacement in a plasma oscillation is $\delta x_i/\delta x_e=- Z\,m_e/m_i$.
   2. Generalize the analysis of Section 1.5 to show that the effective Debye length, $\lambda_D$, of the plasma can be written
      $$\left(\frac{1}{\lambda_D}\right)^2 = \frac{1}{2}\left[\left(\frac{1}{\lambda_{D\,e}}\right)^2+ \left(\frac{1}{\lambda_{D\,i}}\right)^2\right],$$
      where $\lambda_{D\,e} = (\epsilon_0\,T_e/n_e\,e^2)^{1/2}$ and $\lambda_{D\,i} = (\epsilon_0\,T_i/n_i\,Z^{2}\,e^2)^{1/2}$.

2. The perturbed electrostatic potential $\delta{\mit\Phi}$ due to a charge $q$ placed at the origin in a plasma of Debye length $\lambda_D$ is governed by
   $$\left(\nabla^2-\frac{2}{\lambda_D^{2}}\right)\delta{\mit\Phi} = - \frac{q\,\delta({\bf r})}{\epsilon_0}.$$
   Show that the non-homogeneous solution to this equation is
   $$\delta{\mit\Phi}(r) = \frac{q}{4\pi\,\epsilon_0\,r}\,\exp\left(-\frac{\sqrt{2}\,r}{\lambda_D}\right).$$
   Demonstrate that the charge density of the shielding cloud is
   $$\delta\rho(r) = - \frac{2\,q}{4\pi\,r\,\lambda_D^{2}}\exp\left(-\frac{\sqrt{2}\,r}{\lambda_D}\right),$$
   and that the net shielding charge contained within a sphere of radius $r$, centered on the origin, is
   $$Q(r) = -q\left[1-\left(1+\frac{\sqrt{2}\,r}{\lambda_D}\right)\exp\left(-\frac{\sqrt{2}\,r}{\lambda_D}\right)\right].$$

3. A quasi-neutral slab of cold (i.e., $\lambda_D\rightarrow 0$) plasma whose bounding surfaces are normal to the $x$-axis consists of electrons of mass $m_e$, charge $-e$, and mean number density $n_e$, as well as ions of charge $e$, and mean number density $n_e$. The ions can effectively be treated as stationary. The slab is placed in an externally generated, $x$-directed electric field that oscillates sinusoidally at the angular frequency $\omega$. By generalizing the analysis of Section 1.4, show that the relative dielectric constant of the plasma is
   $$\epsilon = 1 - \frac{{\mit\Pi}^{2}}{\omega^2},$$
   where ${\mit\Pi} = (e^2\,n_e/\epsilon_0\,m_e)^{1/2}$.

4. A capacitor consists of two parallel plates of cross-sectional area $A$ and spacing $d\ll \!\sqrt{A}$. The region between the capacitors is filled with a uniform hot plasma of Deybe length $\lambda_D$. By generalizing the analysis of Section 1.5, show that the d.c. capacitance of the device is
   $$C = \frac{\epsilon_0\,A}{d}\frac{(d/\!\sqrt{2}\,\lambda_D)}{\tanh(d/\!\sqrt{2}\,\lambda_D)}.$$

5. A uniform isothermal quasi-neutral plasma with singly-charged ions is placed in a relatively weak gravitational field of acceleration ${\bf g} = -g\,{\bf e}_z$. Assuming, first, that both species are distributed according to the Maxwell-Boltzmann statistics; second, that the perturbed electrostatic potential is a function of $z$ only; and, third, that the electric field is zero at $z=0$ (and well behaved as $z\rightarrow\infty$), demonstrate that the electric field in the region $z>0$ takes the form ${\bf E}=E_z\,{\bf e}_z$, where
   $$E_z(z) = E_0\left[1-\exp\left(\frac{\sqrt{2}\,z}{\lambda_D}\right)\right],$$
   and
   $$E_0 = \frac{m_i\,g}{2\,e}.$$
   Here, $\lambda_D$ is the Debye length, $e$ the magnitude of the electron charge, and $m_i$ the ion mass.

6. Consider a charge sheet of charge density $\sigma$ immersed in a plasma of unperturbed particle number density $n_0$, ion temperature $T_i$, and electron temperature $T_e$. Suppose that the charge sheet coincides with the $y$-$z$ plane. Assuming that the (singly-charged) ions and electrons obey Maxwell-Boltzmann statistics, demonstrate that in the limit $\vert e\,{\mit\Phi}/T_{i,e}\vert\ll 1$ the electrostatic potential takes the form
   $${\mit\Phi}(x) = \frac{\sigma\,\lambda_D}{2\,\epsilon_0}\,{\rm e}^{-\vert x\vert/\lambda_D},$$
   where $\lambda_D =[(\epsilon_0/e^2\,n_0)\,T_i\,T_e/(T_i+T_e)]^{1/2}$.

7. Consider the previous exercise again. Let $T_i=T_e=T$. Suppose, however, that $\vert e\,{\mit\Phi}/T\vert$ is not necessarily much less than unity. Demonstrate that the potential, $V$, of the charge sheet (relative to infinity) satisfies
   $$\frac{e\,V}{T} = \cosh^{-1}\left(1+ \frac{\sigma^2}{16\,\epsilon_0\,n_0\,T}\right).$$
   Furthermore, show that
   $$\tanh(e\,{\mit\Phi}/4\,T)= \tanh(e\,V/4\,T)\,{\rm e}^{-\vert x\vert/\lambda_D},$$
   where $\lambda_D = \sqrt{\epsilon_0\,T/2\,e^2\,n_0}$. Let $x_s$ be the distance from the sheet at which the potential has fallen to $V/{\rm e}$, where $\ln{\rm e}=1$. Sketch $x_s/\lambda_D$ versus $e\,V/T$.

8. A long cylinder of plasma of radius $a$ consists of cold (i.e., $T_i=T_e=0$) singly-charged ions and electrons with uniform number density $n_0$. The cylinder of electrons is perturbed a distance $\delta$ (where $\delta\ll a$) in a direction perpendicular to its axis.
   1. Assuming that the ions are immobile, show that the oscillation frequency of the electron cylinder is
      $${\mit\Pi}=\left(\frac{e^2\,n_0}{2\,\epsilon_0\,m_e}\right)^{1/2},$$
      where $m_e$ is the electron mass.
   2. Assuming that the ions have the finite mass $m_i$, show that the oscillation frequency is
      $${\mit\Pi}=\left[\frac{e^2\,n_0}{2\,\epsilon_0}\left(\frac{1}{m_e}+\frac{1}{m_i}\right)\right]^{1/2}.$$

9. A sphere of plasma of radius $a$ consists of cold (i.e., $T_i=T_e=0$) singly-charged ions and electrons with uniform number density $n_0$. The sphere of electrons is perturbed a distance $\delta$ (where $\delta\ll a$). Assuming that the ions are immobile, show that the oscillation frequency of the electron sphere is
   $${\mit\Pi}=\left(\frac{e^2\,n_0}{3\,\epsilon_0\,m_e}\right)^{1/2},$$
   where $m_e$ is the electron mass.