Home exercise 5.6.4: Two probes

$E_1=5V/m$ and $E_2=5e^{-j120^{\circ }}V/m$. From $E_2=E_1{e}^{-j\beta d}$,

with $\beta >0$ for travel to $+z$. With $k_0=\omega /c=2\pi /3rad/m$, ${\epsilon }_r=(\beta /k_0{)}^2=4(1+3n{)}^2=4,64,196,\dots$

${\epsilon }_r<50$ leaves only $n=0$:

For $1\le {\epsilon }_r\le 100$, $\beta$ lies between ${\beta }_{\min }=k_0$ and ${\beta }_{\max }=10k_0$. The reading is unique when the whole range of $\beta d$ is shorter than $2\pi$:

At $z=d=0.5\mathrm{m}$, $\beta d=2\pi /3=120^{\circ }$. The next candidate, ${\epsilon }_r=64$, would give $\beta d=2\pi /3+2\pi$, the same reading.