Topic: Electromagnetics
An electromagnetic wave propagates through a lossless insulator with a velocity $1.5 \times 10^{10}\text{ cm/s}$. Calculate the relative permittivity ($\epsilon_r$) and relative permeability ($\mu_r$) of the insulator if its intrinsic impedance is $90\pi\text{ ohms}$.
- A. $\epsilon_r = 2.66, \mu_r = 1.5$
- B. $\epsilon_r = 1.5, \mu_r = 2.66$
- C. $\epsilon_r = 1.2, \mu_r = 2.0$
- D. $\epsilon_r = 2.0, \mu_r = 1.2$
Answer: A. $\epsilon_r = 2.66, \mu_r = 1.5$
Explanation: The velocity in a lossless medium is $v = \frac{c}{\sqrt{\mu_r \epsilon_r}}$. Given $v = 1.5 \times 10^8\text{ m/s}$ and $c = 3 \times 10^8\text{ m/s}$, we get $\sqrt{\mu_r \epsilon_r} = 2$. The intrinsic impedance is $\eta = 120\pi \sqrt{\frac{\mu_r}{\epsilon_r}} = 90\pi$. This simplifies to $\sqrt{\frac{\mu_r}{\epsilon_r}} = \frac{3}{4}$, or $\frac{\mu_r}{\epsilon_r} = \frac{9}{16}$. Solving the two equations $\mu_r \epsilon_r = 4$ and $\mu_r = \frac{9}{16} \epsilon_r$, we get $\epsilon_r^2 = 64/9 \Rightarrow \epsilon_r = 2.66$ and $\mu_r = 1.5$.