Question

Ignoring reflection at the air-water boundary, if the amplitude of a 10 GHz incident wave in air is 20 V/m at the water surface, at what depth will it be down to 1 µV/m? Water is characterized by er = 81, µr = 1, and σ = 0.1 S/m. Can water described above, at 10 GHz, be described as a low-loss dielectric, good conductor, or "in-between"?

Answer 1

Answer:

0.80267 m

Explanation:

E(z) = Electric field = 1 µV/m

$E_0$ = 20 V/m

z = Depth

$\sigma$ = Conductivity = 0.1 S/m

$\epsilon_r$ = 81

$\mu$ = Impedance of free space = $120\pi\ \Omega$

Frequency is given by

$E(z)=E_0e^{-\alpha z}$

Parameter is given by

$\alpha=\dfrac{\sigma}{2}\sqrt{\dfrac{\mu}{\epsilon_r}}\\\Rightarrow \alpha=\dfrac{1}{2}\sqrt{\dfrac{(120\pi)^2}{81}}\\\Rightarrow \alpha=20.94395\ N_p/m$

From the first equation

$1\times 10^{-6}=20e^{-20.94395z}\\\Rightarrow ln\dfrac{1\times 10^{-6}}{20}=-20.94395z\\\Rightarrow z=\dfrac{ln\dfrac{1\times 10^{-6}}{20}}{-20.94395}\\\Rightarrow z=0.80267\ m$

The depth is 0.80267 m