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.
(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.
To solve this problem, we can use the Fresnel equations for reflection and transmission at the interface between two media. Let's go step by step:
a) Coefficients for reflection (R) and transmission (T):
The equations for reflection and transmission coefficients are given by:
R = (n1 - n2) / (n1 + n2)
T = 2 * n1 / (n1 + n2)
Where n1 and n2 are the refractive indices of the two media.
In this case, since we are considering a lossless nonmagnetic medium with Er = 9 (relative permittivity), the refractive index can be calculated using the formula:
n = sqrt(Er)
n2 is the refractive index in the second medium, which is free space (n2 = 1).
n1 = sqrt(Er) = sqrt(9) = 3
Now we can calculate the reflection (R) and transmission (T) coefficients:
R = (n1 - n2) / (n1 + n2) = (3 - 1) / (3 + 1) = 2 / 4 = 1/2 = 0.5
T = 2 * n1 / (n1 + n2) = 2 * 3 / (3 + 1) = 6 / 4 = 3/2 = 1.5
So, the coefficients for reflection (R) and transmission (T) are R = 0.5 and T = 1.5.
b) Standing-wave ratio in free-space (SWR, S):
The standing-wave ratio (SWR) is defined as the ratio of the maximum electric field amplitude to the minimum electric field amplitude in the wave. In this case, the SWR in free space can be calculated using the reflection coefficient (R):
S = (1 + |R|) / (1 - |R|) = (1 + 0.5) / (1 - 0.5) = 1.5 / 0.5 = 3
So, the standing-wave ratio (SWR) in free space is 3.
c) Average power densities:
The average power density for a wave can be calculated using the formula:
P = (1/2) * (E * H*),
Where E is the electric field amplitude and H is the magnetic field amplitude.
For transmitted wave:
STU = (1/2) * (|T| * H)^2
For reflected wave:
SAU = (1/2) * (|R| * H)^2
For incident wave:
sau = (1/2) * (E * H)^2
In this case, the amplitude of the electric field (E) is given as 20 V/m, and the magnetic field amplitude (H) is given as 1.
STU = (1/2) * (|T| * H)^2 = (1/2) * (1.5 * 1)^2 = (1/2) * (1.5^2) = (1/2) * 2.25 = 1.125
SAU = (1/2) * (|R| * H)^2 = (1/2) * (0.5 * 1)^2 = (1/2) * (0.5^2) = (1/2) * 0.25 = 0.125
sau = (1/2) * (E * H)^2 = (1/2) * (20 * 1)^2 = (1/2) * (20^2) = (1/2) * 400 = 200
So, the average power densities for the transmitted wave (STU), reflected wave (SAU), and incident wave (sau) are 1.125 W/m^2, 0.125 W/m^2, and 200 W/m^2, respectively.