PX284 - R1 - waves in dielectrics

maxwell's equations in matter with no sources, ie: $ρ_{f} - {\vec{J}}_{f} = 0$ , for linear, isotropic materials

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

this is a wave equation:

\nabla^{2} \vec{E} = \frac{1}{v^{2}} \frac{\partial^{2} \vec{E}}{\partial t^{2}}

the wave speed:

v = \frac{1}{\sqrt{μ_{0} ε_{0} μ_{r} ε_{r}}} = \frac{c}{\sqrt{μ_{r} ε_{r}}} = \frac{c}{n}

$n = \sqrt{μ_{r} ε_{r}}$ is called the refractive index
for most transparent matter (allowing propagation of EM waves), $μ_{r} ≃ 1$ , so $n ≃ \sqrt{ε_{r}}$
$ϵ_{r}$ is frequency dependent
plane waves in matter (linear, isotropic response)

\begin{matrix} \vec{E} = E_{0} \exp (i (\vec{k} \cdot \vec{x} - ω t)) \hat{E} \\ \vec{H} = H_{0} \exp (i (\vec{k} \cdot \vec{x} - ω t)) \hat{H} \\ \vec{\nabla} ⟹ i \vec{k} \\ \partial_{t} ⟹ - i ω \end{matrix}

maxwell's equations with no free charges or currents :

\begin{matrix} \vec{k} \cdot \vec{D} = 0 \\ \vec{k} \cdot \vec{B} = 0 \\ \vec{k} \times \vec{E} = ω \vec{B} \\ \vec{k} \times \vec{H} = - ω \vec{D} \end{matrix}

as in free space: $\vec{k} ⊥ \vec{E}, \vec{H}$ and $\vec{E} ⊥ \vec{H}$

E_{0} = \frac{ω B_{0}}{| \vec{k} |} = v B_{0} = v μ_{0} μ_{r} H_{0}

defining the impedance:

Z = \frac{E_{0}}{H_{0}} = v μ_{0} μ_{r} = \sqrt{\frac{μ_{0} μ_{r}}{ε_{0} ε_{r}}} = Z_{0} \sqrt{\frac{μ_{r}}{ε_{r}}}

where, $Z_{0} ≃ 376.73 Ω$ is the impedance of free space

in materials with $μ_{r} ≃ 1 :$

Z ≃ \frac{Z_{0}}{\sqrt{ε_{r}}} ≃ \frac{Z_{0}}{n}