A3 - electromagnetic waves

starting with Faraday's equation, and taking its curl:

\begin{matrix} \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) + \vec{\nabla} \times \frac{\partial \vec{B}}{\partial t} = 0 \\ \vec{\nabla} (\vec{\nabla} \cdot \vec{E}) - \nabla^{2} \vec{E} + \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B}) = 0 \end{matrix}

in free space, $θ = 0, \vec{j} = 0 :$

\begin{matrix} \vec{\nabla} \cdot \vec{E} = 0 \\ \vec{\nabla} \times \vec{B} = μ_{0} ε_{0} \frac{\partial \vec{E}}{\partial t} \\ ∴ μ_{0} ε_{0} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = \nabla^{2} \vec{E} \end{matrix}

this is the wave equation for light, with the speed of light $c = 1 / \sqrt{μ_{0} ε_{0}}$
in free space, the solution is a plane wave:

\begin{matrix} \vec{E} (\vec{r}, t) = {\vec{E}}_{0} \cdot \exp [i (ω t - \vec{k} \cdot \vec{r})] \\ \vec{B} (\vec{r}, t) = {\vec{B}}_{0} \cdot \exp [i (ω t - \vec{k} \cdot \vec{r})] \end{matrix}

the phase velocity, $| {\vec{v}}_{p} | = c = ω / | \vec{k} |$
solenoidal condition:

\vec{\nabla} \cdot \vec{B} = - i \vec{k} \cdot \vec{B}

so in calculations for plane waves, can replace $\vec{\nabla}$ with $- i \vec{k}$ and $\frac{\partial}{\partial t}$ with $i ω$

\vec{\nabla} \cdot \vec{E} = 0 ⟹ \vec{k} \cdot \vec{E} = 0

$\vec{k}$ is perpendicular to both $\vec{E}$ and $\vec{B}$
hence, plane waves are transverse (TEM)
Faraday's law:

\begin{matrix} \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \\ - i \vec{k} \times \vec{E} = - i ω \vec{B} \\ \vec{B} = \frac{\vec{k} \times \vec{E}}{ω} \\ | B | = \frac{| E |}{c} or E_{0} = c B_{0} \end{matrix}

power flow in a wave:

\vec{S} = \vec{E} \times \vec{H}

for a plane wave in free spave:

\vec{S} = \frac{\vec{E} \times \vec{B}}{μ_{0}} = \frac{| E | | B |}{μ_{0}} \hat{k} = \frac{E_{0}^{2}}{μ_{0} c} \hat{k}

$ε_{0} μ_{0} c^{2} = 1 :$
define the impedance:

\begin{matrix} Z = \frac{| E |}{| H |} = \frac{E_{0}}{B_{0}} μ_{0} μ_{r} \\ Z = \frac{ω}{| k |} μ_{0} μ_{r} = \sqrt{\frac{μ_{0} μ_{r}}{ε_{0} ε_{r}}} = Z_{0} \sqrt{\frac{μ_{r}}{ε_{r}}} \end{matrix}

$Z_{0} = \sqrt{μ_{0} / ε_{0}}$ is the impedance of free space

\vec{S} = \frac{E_{0}^{2}}{Z_{0}} \hat{k}