Home exercise 5.1.1: A plane wave in free space

$\nabla \cdot D={\epsilon }_0\partial E_x/\partial x=0$ and $\nabla \cdot B={\mu }_0\partial H_y/\partial y=0$, because $E_x$ does not depend on $x$ and $H_y$ does not depend on $y$, consistent with $\rho =0$.

For a field with only an $x$ component depending on $z$,

They are equal if $k={\mu }_0\omega /{\eta }_0$, that is $\omega /k={\eta }_0/{\mu }_0=\sqrt{{\mu }_0/{\epsilon }_0}/{\mu }_0=1/\sqrt{{\mu }_0{\epsilon }_0}=c$.

For a field with only a $y$ component depending on $z$,

With $J=0$ they are equal if $k/{\eta }_0={\epsilon }_0\omega$, that is $\omega /k=1/({\eta }_0{\epsilon }_0)=1/\sqrt{{\mu }_0{\epsilon }_0}=c$, the same condition.

With ${\mu }_0=4\pi \times 10^{-7}H/m$ and ${\epsilon }_0=10^{-9}/(36\pi )F/m$ the product ${\mu }_0{\epsilon }_0=10^{-16}/9{\mathrm{s}}^2/{\mathrm{m}}^2$ and the ratio ${\mu }_0/{\epsilon }_0=(120\pi {)}^2{\Omega }^2$, so $c=3\times 10^{8}m/s$ and ${\eta }_0=120\pi \Omega$ exactly. The pair $(E,H)$ is a wave travelling towards $+z$ with speed $\omega /k=c$: at a fixed phase $u$ one has $z=(\omega t-u)/k$, which advances at $\omega /k$. $E$, $H$ and the direction of travel $a_z$ form a right-handed triad, $E\times H\propto a_z$. It is the $\partial D/\partial t$ term that closes the loop: without it Ampère's law would give $\nabla \times H=0$ and no wave could exist in empty space. Maxwell's identification of $1/\sqrt{{\mu }_0{\epsilon }_0}$, obtained from purely electrical measurements, with the measured speed of light was the first evidence that light is an electromagnetic wave.