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A plane wave with a frequency of $3 \mathrm{Gl} z$ is propagating in an unbounded material with $\epsilon_r=7$ and $\mu_r=3$. Compute the wavelength. phase velocity, and wave impedance for this wave.
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Solution:

From the phase velocity is,

$$v_p=\frac{1}{\sqrt{\mu \epsilon}}=\frac{c}{\sqrt{\mu_r \epsilon_r}}=\frac{3 \times 10^8}{\sqrt{(7)(3)}}=6.55 \times 10^7 \mathrm{~m} / \mathrm{sec} .$$

This is slower than the speed of light in free space by a factor of $\sqrt{21}=4.58$. From (1.48) the wavelength is,

$$\lambda=\frac{v_p}{f}=\frac{6.55 \times 10^7}{3 \times 10^9}=0.0218 \mathrm{~m} .$$

The wave impedance is,

$$\eta=\sqrt{\frac{\mu}{\epsilon}}=\eta_0 \sqrt{\frac{\mu_r}{\epsilon_r}}=377 \sqrt{\frac{3}{7}}=246.8 \Omega .$$
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