Answer to Question #93560 in Electricity and Magnetism for DD

Question

A uniform plane wave of 10 kHz travelling in free space strikes a large block of a
material having ε = 9ε0, μ = 4μ0 and σ = 0 normal to the surface. If the incident
magnetic field vector is given by
B =10^− 3sin (ωt −βy) kˆ
tesla
Write the complete expressions for the incident, reflected, and transmitted field vectors.

Answer 1

We have frequency $f=10\cdot10^3\text{ Hz}.$ First, calculate required parameters for each medium (medium of free space has index 0, medium within the block of some material has index 2). Reflected wave has index $r$, index $t$ is for transmitted wave.

$$\beta_0=2\pi f\sqrt{\mu_0\epsilon_0}=\frac{2\pi f}{c}=\frac{\pi}{15000}\text{ m}^{-1},\\ \beta_{r}=-\beta_0,\\ \beta_t=2\pi f\sqrt{\mu_2\epsilon_2}=\frac{\pi}{2500}\text{ m}^{-1},\\ \eta_0=\sqrt{\frac{\mu_0}{\epsilon_0}}=120\pi\space\Omega,\\ \eta_2=\sqrt{\frac{4\mu_0}{9\epsilon_0}}=80\pi\space\Omega,\\ r=\frac{\eta_2-\eta_0}{\eta_2+\eta_0}=-0.2,\\ t=1+r=0.8.$$

Magnitudes of the incident E-field and H-field:

$$E_0=\frac{B_0}{\sqrt{\mu_0\epsilon_0}}=3\cdot10^5\text{ V/m}.$$

The equations for incident wave:

$$\textbf{E}_i=E_oe^{-j\beta_0 z}\hat\textbf{x},\\ \space\\ \textbf{H}_i=\frac{E_o}{\eta_0}e^{-j\beta_0 z}\hat\textbf{y}.$$

$$\textbf{E}_i =3\cdot10^5\cdot\text{sin}\Big(2\pi\cdot10^4 t-\frac{\pi}{15000}y\Big)(-\hat\textbf{x})\space\frac{\text{V}}{\text{m}},\\ \space\\ \textbf{H}_i=\frac{\textbf{B}_i}{\mu_0}=\frac{2500}{\pi}\cdot\text{sin}\Big(2\pi\cdot10^4 t-\frac{\pi}{15000}y\Big)\hat\textbf{z}\space\frac{\text{V}}{\text{m}}.$$

The equations for reflected wave with reflection coefficient $r$:

$$\textbf{E}_r=rE_oe^{-j\beta_r z}\hat\textbf{x},\\ \space\\ \textbf{H}_r=-r \frac{E_o}{\eta_0}e^{-j\beta_r z}\hat\textbf{y}.$$

$$\textbf{E}_r=-6\cdot10^4\cdot\text{sin}\Big(2\pi\cdot10^4 t+\frac{\pi}{15000}y\Big)\hat\textbf{x}\space\frac{\text{V}}{\text{m}},\\ \space\\ \textbf{H}_r=\frac{500}{\pi}\cdot\text{sin}\Big(2\pi\cdot10^4 t+\frac{\pi}{15000}y\Big)\hat\textbf{z}\space\frac{\text{A}}{\text{m}}.$$

The equations of transmitted wave with transmission coefficient $t$:

$$\textbf{E}_t=t E_oe^{-\beta_2 z}\hat\textbf{x},\\ \space\\ \textbf{H}_t=t\frac{E_o}{\eta_2}e^{-j\beta_2 z}\hat\textbf{y}.$$

$$\textbf{E}_t=24\cdot10^4\cdot\text{sin}\Big(2\pi\cdot10^4 t-\frac{\pi}{2500}y\Big)(-\hat\textbf{x})\space\frac{\text{V}}{\text{m}},\\ \space\\ \textbf{H}_\tau=\frac{2000}{\pi}\cdot\text{sin}\Big(2\pi\cdot10^4 t-\frac{\pi}{2500}y\Big)\hat\textbf{z}\space\frac{\text{A}}{\text{m}}.$$