(a) The coefficients for reflection and transmission are given by:
$$
R_\perp = \frac{Z_2\cos\theta_i - Z_1\cos\theta_t}{Z_2\cos\theta_i + Z_1\cos\theta_t} = \frac{9\cos(0) - 377\cos(\theta_t)}{9\cos(0) + 377\cos(\theta_t)} \approx -0.996
$$
$$
T_\perp = \frac{2Z_2\cos\theta_i}{Z_2\cos\theta_i + Z_1\cos\theta_t} = \frac{2\cdot 9\cos(0)}{9\cos(0) + 377\cos(\theta_t)} \approx 0.008
$$
where $\theta_i$ is the angle of incidence (0 in this case), $\theta_t$ is the angle of transmission (given by Snell's law), $Z_1$ is the impedance of free-space (377 ohms), and $Z_2$ is the impedance of the medium (given by $\eta / \sqrt{\mu_r \epsilon_r}$, where $\eta$ is the impedance of free-space and $\mu_r = 1$ since the medium is nonmagnetic).
(b) The standing wave ratio (SWR) in free-space is given by:
$$
S = \frac{1 + |R_\perp|}{1 - |R_\perp|} \approx 324
$$
(c) The average power densities for the transmitted (STU), reflected (SAU), and incident (SAI) waves are given by:
$$
STU = \frac{1}{2} \sqrt{\frac{\mu_0}{\epsilon_0}} |E_t|^2 = \frac{1}{2} \sqrt{\frac{\mu_0}{\epsilon_0}} |T_\perp|^2 |E_i|^2 \approx 2.79 \times 10^{-6} \text{ W/m}^2
$$
$$
SAU = \frac{1}{2} \sqrt{\frac{\mu_0}{\epsilon_0}} |E_r|^2 = \frac{1}{2} \sqrt{\frac{\mu_0}{\epsilon_0}} |R_\perp|^2 |E_i|^2 \approx 3.94 \text{ W/m}^2
$$
$$
SAI = \frac{1}{2} \sqrt{\frac{\mu_0}{\epsilon_0}} |E_i|^2 = \frac{1}{2} \sqrt{\frac{\mu_0}{\epsilon_0}} |E_i|^2 = 2.52 \times 10^{-5} \text{ W/m}^2
$$
where $E_i$ is the amplitude of the incident field, $E_r$ is the amplitude of the reflected field, and $E_t$ is the amplitude of the transmitted field. Since the medium is lossless, there is no attenuation of the wave as it passes through, so the transmitted power density is simply proportional to the square of the transmission coefficient. The reflected power density is proportional to the square of the reflection coefficient, and is much larger than the incident and transmitted power densities due to the high reflection coefficient.
The Ē of a plane wave in free-space has an amplitude of 20 V/m. This plane wave hits normally the surface of a lossless nonmagnetic medium which has Er = 9, H = 1. (a) Find the coefficients for reflection (T) and transmission (T), (b) the standing-wave ratio in the free-space (SWR, S), (c) and the average power densities of the transmit- ted (STU), reflected (SAU), and incident (sau) waves.
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