6.2.9 Problems and solutions

The problems of this subsection use the plane wave in a medium $(\epsilon ,\mu ,\sigma )$, $E=a_x{E}_0{e}^{-\gamma z}$ (time factor $e^{\mathrm{j}\omega t}$ suppressed), with

the low-loss forms $\alpha \approx \frac{\sigma }{2}\sqrt{\mu /\epsilon }$, $\beta \approx \omega \sqrt{\mu \epsilon }$, ${\eta }_c\approx \sqrt{\mu /\epsilon }(1+\mathrm{j}p/2)$ for $p\ll 1$, the good-conductor forms $\alpha =\beta =1/\delta =\sqrt{\pi f\mu \sigma }$, ${\eta }_c=(1+\mathrm{j})/\sigma \delta$, $R_s=1/\sigma \delta$ for $p\gg 1$, and for a collisionless plasma ${\epsilon }_p={\epsilon }_0(1-f_p^2/f^2)$ with $f_p\approx 8.9836\sqrt{N}\mathrm{Hz}$, $v_p{v}_g=c^2$. The constants are those of this course, ${\epsilon }_0=10^{-9}/36\pi F/m$, ${\mu }_0=4\pi \times 10^{-7}H/m$, $c=3\times 10^{8}m/s$, ${\eta }_0=120\pi \Omega$, $e=1.602\times 10^{-19}\mathrm{C}$, $m=9.11\times 10^{-31}\mathrm{kg}$, and a field factor $e^{-\alpha z}$ is $8.686\alpha z\mathrm{dB}$. Problems 6.2.1–6.2.4 come from the lecture on propagation in different media, Problems 6.2.5–6.2.8 from the exercise lecture.