Exercise 5.5.1: A Gaussian pulse in a dielectric

$5z+10^{9}t=5(z+2\times 10^{8}t)$. The field is a function of $z+vt$ with $v=2\times 10^{8}m/s$, so the pulse travels in the $-z$ direction. For a nonmagnetic medium $v=c/\sqrt{{\epsilon }_r}$:

The peak sits where the argument is zero, $z=-vt=-2\times 10^8\times 20\times 10^{-9}=-4\mathrm{m}$.

At a fixed $z$ the field is $10e^{-25(z+vt{)}^2}$, and $e^{-25s^2}=\frac{1}{2}$ for $|s|=\sqrt{\ln 2}/5=0.1665\mathrm{m}$. The pulse is above half its peak over a length of $0.333\mathrm{m}$ (the full width at half maximum), and this length passes a fixed point in

Write $E_x=F(z+vt)$. Then $\nabla \times E=a_y\partial E_x/\partial z=a_y{F}^{\prime }$ and Faraday's law gives ${\mu }_0\partial H_y/\partial t=-F^{\prime }$. Since $\partial F/\partial t=v{F}^{\prime }$, this is $H_y=-F/({\mu }_{0}v)$ (no static part), and ${\mu }_{0}v=\sqrt{{\mu }_0/\epsilon }=\eta =120\pi /1.5=251\Omega$:

Units: ${\mu }_{0}v$ is $(H/m)(m/s)=\Omega$. For a wave towards $-z$ the magnetic field has the opposite sign to the one towards $+z$: $E\times H=-(E_x^2/\eta )a_z$ points in the direction of travel.