Exercises

1. A general electromagnetic wave pulse propagating in the $z$ -direction at velocity $u$ is written

   $${\bf E} = P(z-u\,t)\,{\bf e}_x + Q(z-u\,t)\,{\bf e}_y+ R(z-u\,t)\,{\bf e}_z,$$

   $${\bf B} = \frac{S(z-u\,t)}{u}\,{\bf e}_x + \frac{T(z-u\,t)}{u}\,{\bf e}_y+ \frac{U(z-u\,t)}{u}\,{\bf e}_z,$$

   
   
   where $P$ , $Q$ , $R$ , $S$ , $T$ , and $U$ are arbitrary functions. In order to exclude electrostatic and magnetostatic fields, these functions are subject to the constraint that $\langle P\rangle=\langle Q\rangle=\langle R\rangle=\langle S\rangle=\langle T\rangle=\langle U\rangle =0$ , where
   $$\langle P\rangle = \int_{-\infty}^{\infty}P(x)\,dx.$$
   Suppose that the pulse propagates through a uniform dielectric medium of dielectric constant $\epsilon$ . Demonstrate from Maxwell's equation that $u = c/\sqrt{\epsilon}$ , $R=U=0$ , $S=-Q$ , and $T=P$ . Incidentally, this result implies that a general wave pulse is characterized by two arbitrary functions, corresponding to the two possible independent polarizations of the pulse.

2. Show that the mean energy flux due to an electromagnetic wave of angular frequency $\omega$ propagating though a good conductor of conductivity $\sigma$ can be written
   $$\langle I\rangle = \frac{E^{\,2}}{\sqrt{8}\,Z},$$
   where $E$ is the peak electric field-strength, and $Z=(\epsilon_0\,\omega/\sigma)^{1/2}$ .

3. Consider an electromagnetic wave propagating in the positive $z$ -direction through a good conductor of conductivity $\sigma$ . Suppose that the wave electric field is
   $$E_x(z,t) = E_0\,{\rm e}^{-z/d}\,\cos(\omega\,t-z/d),$$
   where $d$ is the skin-depth. Demonstrate that the mean electromagnetic energy flux across the plane $z=0$ matches the mean rate at which electromagnetic energy is dissipated per unit area due to Joule heating in the region $z>0$ .

4. A plane electromagnetic wave, linearly polarized in the $x$ -direction, and propagating in the $z$ -direction through an electrical conducting medium of conductivity $\sigma$ and relative dielectric constant unity, is governed by

   $$\frac{\partial H_y}{\partial t} = - \frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z},$$

   $$\frac{\partial E_x}{\partial t} =-\frac{\sigma}{\epsilon_0}\,E_x -\frac{1}{\epsilon_0}\,\frac{\partial H_y}{\partial z},$$

   
   
   where $E_x(z,t)$ and $H_y(z,t)$ are the electric and magnetic components of the wave. Derive an energy conservation equation of the form
   $$\frac{\partial{\cal E}}{\partial t} + \frac{\partial {\cal I}}{\partial z} =- \sigma\,E_x^{\,2},$$
   where ${\cal E}$ is the electromagnetic energy per unit volume, and ${\cal I}$ the electromagnetic energy flux. Give expressions for ${\cal E}$ and ${\cal I}$ . What does the right-hand side of the previous equation represent? Demonstrate that $E_x$ obeys the wave-diffusion equation
   $$\frac{\partial^{\,2} E_x}{\partial t^{\,2}} + \frac{\sigma}{\epsi... ...artial E_x}{\partial t}= c^{\,2}\,\frac{\partial^{\,2} E_x}{\partial z^{\,2}},$$
   where $c=1/\sqrt{\epsilon_0\,\mu_0}$ . Consider the high frequency, low conductivity, limit $\omega\gg \sigma/\epsilon_0$ . Show that a wave propagating into the medium varies as

   $$E_x(z,t) \simeq E_0\,\cos[k\,(c\,t-z)]\,{\rm e}^{-z/\delta},$$

   $$H_y(z,t) \simeq Z_0^{\,-1}\,E_0\,\cos[k\,(c\,t-z)-1/(k\,\delta)]\,{\rm e}^{-z/\delta},$$

   
   
   where $k=\omega/c$ , $\delta = 2/(Z_0\,\sigma)$ , and $Z_0=\sqrt{\mu_0/\epsilon_0}$ . Demonstrate that $k\,\delta \ll 1$ : that is, the wave penetrates many wavelengths into the medium.

5. Consider a uniform plasma of plasma frequency $\omega_p$ containing a uniform magnetic field $B_0\,{\bf e}_z$ . Show that left-hand circularly polarized electromagnetic waves can only propagate parallel to the magnetic field provided that $\omega > -{\mit\Omega}/2 + \sqrt{{\mit\Omega}^{\,2}/4+\omega_p^{\,2}}$ , where ${\mit\Omega}=e\,B_0/m_e$ is the electron cyclotron frequency. Demonstrate that right-hand circularly polarized electromagnetic waves can only propagate parallel to the magnetic field provided that their frequencies do not lie in the range ${\mit\Omega}\leq \omega\leq{\mit \Omega}/2 + \sqrt{{\mit\Omega}^{\,2}/4+\omega_p^{\,2}}$ .