PX284 - S3b - reflections on boundaries with metals

considering normal incidence, ie: $i = r = t = 0$ , and p-polarization:

\begin{matrix} H_{i} - H_{r} = H_{t} \\ E_{i} + E_{r} = E_{t} \end{matrix}

in the limit of perfect conductor, $g \to \infty :$

\begin{matrix} E_{t} = 0 \\ E_{r} = - E_{i} \\ r = \frac{E_{r}}{E_{i}} = - 1 \\ t = \frac{E_{t}}{E_{i}} = 0 \end{matrix}

for good conductors:

\begin{matrix} \frac{E_{i}}{Z_{1}} - \frac{E_{r}}{Z_{1}} = \frac{E_{t}}{Z_{2}} \\ E_{i} + E_{r} = E_{t} = \frac{Z_{2}}{Z_{1}} (E_{i} - E_{r}) \\ (1 - \frac{Z_{2}}{Z_{1}}) E_{i} = (1 + \frac{Z_{2}}{Z_{1}}) E_{r} \\ r = \frac{E_{r}}{E_{i}} = \frac{Z_{2} - Z_{1}}{Z_{2} + Z_{1}} \end{matrix}

for air-metal boundary: $Z_{1} \approx Z_{0} \approx 377 ω$

Z_{2} = Z_{c} = \frac{μ ω δ}{2} (1 - i) = \sqrt{\frac{μ ω}{2 g}} (1 - i)

considering $ω = 2 π \times 2 \times 10^{6}$ rad s $^{- 1}$

| Z_{c} | ≃ 7.3 \times 10^{4} Ω

\frac{Z_{C}}{Z_{0}} ≃ 10^{- 6}

r = - \frac{1 - \frac{Z_{C}}{Z_{0}}}{1 + \frac{Z_{C}}{Z_{0}}} ≃ (1 - \frac{2 Z_{c}}{Z_{0}})

where, $1$ is the perfect conductor limit, and the additional term is the small correction