Exercises

1. Given that $\rho$$= \rho\,(-\cos\gamma\,{\bf e}_1+ \sin\gamma\,{\bf e}_2)$, and ${\bf u} = {\mit\Omega}\,$$\rho$$\times {\bf b}$, where $\rho = u_\perp/{\mit\Omega}$, and ${\bf e}_1$, ${\bf e}_2$, ${\bf b} \equiv {\bf B}/B$ are a right-handed set of mutually perpendicular unit basis vectors, demonstrate that:

   1. $$\langle$$   $\rho$$$\,$$$\rho$$$\rangle = \frac{u_\perp^{2}}{2\,{\mit\Omega}^{2}}\,\left({\bf I}- {\bf b}\,{\bf b}\right).$$

   2. $$e\,\langle {\bf u}\times ($$$\rho$$$\cdot\nabla)\,{\bf B}\rangle =- \mu\,\nabla B.$$

   3. $$e\,\langle {\bf u}\cdot($$$\rho$$$\cdot\nabla)\,{\bf E}\rangle = \mu\,\frac{\partial B}{\partial t}.$$

   4. $$e\,\langle {\bf u}\cdot($$$\rho$$$\cdot\nabla)\,{\bf A}\rangle = - \mu\,B.$$

   Here, $\mu=m\,u_\perp^{2}/(2\,B)$, and $\langle\cdots\rangle\equiv \oint(\cdots)\,d\gamma/2\pi$.

2. 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 mass $m_i$, charge $e$, and mean number density $n_e$. The slab is fully magnetized by a uniform $y$-directed magnetic field of magnitude $B$. The slab is then subject to an externally generated, uniform, $x$-directed electric field that is gradually ramped up to a final magnitude $E_0$. Show that, as a consequence of ion polarization drift, the final magnitude of the electric field inside the plasma is
   $$E_1 \simeq \frac{E_0}{\epsilon},$$
   where
   $$\epsilon = 1 +\frac{c^2}{V_A^{2}},$$
   and $V_A = B/\!\sqrt{\mu_0\,n_e\,m_i}$ is the so-called Alfvén velocity.

3. A linear magnetic dipole consists of two infinite straight wires running parallel to the $z$-axis. The first wire lies at $x=0$, $y=d/2$ and carries a steady current $I$. The second lies at $x=0$, $y=-d/2$ and carries a steady current $-I$. Let $r=(x^2+y^2)^{1/2}$ and $\theta=\tan^{-1}(y/x)$. Demonstrate that the magnetic field generated by the dipole in the region $r\gg d$ can be written
   $${\bf B} = \nabla\psi\times {\bf e}_z,$$
   where
   $$\psi = \frac{\mu_0\,I\,d}{2\pi}\,\frac{\sin\theta}{r}.$$

4. Consider a particle of charge $e$, mass $m$, and energy ${\cal E}$, trapped on a field-line of the linear magnetic dipole discussed in the previous exercise. Let $\vartheta = \pi/2 - \theta$. Suppose that the field-line crosses the “equatorial” plane $\vartheta=0$ at $r=r_{\rm eq}\gg d/2$, and that the magnetic field-strength at this point is $B_{\rm eq}$. Suppose that the particle's mirror points lie at $\vartheta=\pm\vartheta_m$. Assume that the particle's gyroradius is much smaller than $r_{\rm eq}$, and that the electric field-strength is negligible.
   1. Demonstrate that the variation of the particle's perpendicular and parallel velocity components with the “latitude” $\vartheta$ is

      $$v_\perp = \left(\frac{2\,{\cal E}}{m}\right)^{1/2}\frac{\cos\vartheta_m}{\cos\vartheta},$$

      $$v_\parallel =\pm \left(\frac{2\,{\cal E}}{m}\right)^{1/2}\left(1-\frac{\cos^2\vartheta_m}{\cos^2\vartheta}\right)^{1/2},$$

      respectively.
   2. Demonstrate that the particle's bounce period is
      $$\tau_b =\frac{ \sqrt{2}\,\pi\,r_{\rm eq}}{({\cal E}/m)^{1/2}}.$$

   3. Demonstrate that the particle drifts in the $z$-direction with the mean velocity
      $$\langle v_{d}\rangle = \frac{2\,{\cal E}}{e\,B_{\rm eq}\,r_{\rm eq}}.$$

5. A charged particle of mass $m$ is trapped in a static magnetic mirror field given by
   $$B_z = B_0\left(1+\frac{z^2}{L^2}\right),$$
   and has total kinetic energy ${\cal E}$, and pitch angle $\alpha$ at $z=0$. Assuming that the electric field is negligible, and that the particle's gyroradius is much less than $L$, use guiding center theory to show that the bounce time is
   $$\tau_b = \frac{2\pi\,L}{\sin\alpha\sqrt{2\,{\cal E}/m}}.$$

6. A particle of charge $e$, mass $m$, and energy ${\cal E}$, is trapped in a one-dimensional magnetic well of the form
   $$B(x,t) = B_0\,(1+k^2\,x^2),$$
   where $B_0$ is constant, and $k(t)$ is a very slowly increasing function of time. Suppose that the particle's mirror points lie at $x=\pm x_m(t)$, and that its bounce time is $\tau_b(t)$. Demonstrate that, as a consequence of the conservation of the first and second adiabatic invariants,

   $$x_m(t) = x_m(0)\left[\frac{k(0)}{k(t)}\right]^{1/2},$$

   $$\tau_b(t) =\tau_b(0)\left[\frac{k(0)}{k(t)}\right],$$

   $${\cal E}(t) = {\cal E}_{0\,\perp} + \left[\frac{k(t)}{k(0)}\right]{\cal E}_{0\,\parallel}.$$

   Here, ${\cal E}_{0\,\perp}$ is the perpendicular energy [i.e., $(1/2)\,m\,v_\perp^{2}$], and ${\cal E}_{0\,\parallel}$ is the parallel energy [i.e., $(1/2)\,m\,v_\parallel^{2}$], both evaluated at $x=0$ and $t=0$. Assume that the particle's gyroradius is relatively small, and that the electric field-strength is negligible.

7. Consider the static magnetic field
   $$B_z(y) = \left\{ \begin{array}{lll} B_0&\mbox{\hspace{0.5cm}... ...a)&&\vert y\vert< a\\ [0.5ex] -B_0 &&y < -a \end{array}\right.$$
   which corresponds to a current sheet such as that found in the Earth's magnetotail. Let the electric field be negligible. Consider the orbits of charged particles of mass $m$ and charge $e$ whose gyroradii, $\rho$, are not necessarily much smaller than the shear-length, $a$, of the magnetic field. In this situation, guiding center theory is inapplicable. The particles' orbits can only be analyzed by directly solving their equations of perpendicular motion. It is easily demonstrated that some orbits do not cross the neutral plane ($y=0$) and resemble conventional magnetized particle orbits, whereas others meander across the neutral plane and are quite different from conventional orbits.
   1. Consider a particle orbit that does not cross the neutral plane, but is instead confined to the region $y_+\geq y\geq y_-$, where $a>y_+>y_->0$. Demonstrate that the mean drift velocity of the particle in the $x$-direction can be written
      $$\langle v_x\rangle = - \left(\frac{{\mit\Omega}_0}{4\,a}\right)(y_+^{2}+y_-^{2})\,(1-\alpha),$$
      where ${\mit\Omega}_0=e\,B_0/m$, and
      $$\alpha = \frac{\int_{-1}^1 (1+\kappa\,\zeta)^{1/2}\,(1-\zeta^{2})... ...\,d\zeta}{\int_{-1}^1 (1+\kappa\,\zeta)^{-1/2}\,(1-\zeta^{2})^{-1/2}\,d\zeta},$$
      with $\kappa=(y_+^{2}-y_-^{2})/(y_+^{2}+y_-^{2})$. Show that in the limit $\vert y_+-y_-\vert/a\ll 1$ the previous result is consistent with that obtained from conventional guiding center theory.
   2. Consider a particle orbit that is confined to the region $y_0\geq y\geq -y_0$, where $a>y_0$, and is such that $v_x=0$ when $y=0$. Demonstrate that the mean drift velocity in the $x$-direction is
      $$\langle v_x\rangle =+ 0.223\left(\frac{{\mit\Omega}_0}{a}\right)y_0^{2}.$$