A microwave antenna radiating at 10 GHz is to be protected from the environment by a plastic shield of dielectric constant 2.5. What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9.199.]
T^{-1}=\frac{1}{4 n_{1} n_{3}}\left[\left(n_{1}+n_{3}\right)^{2}+\frac{\left(n_{1}^{2}-n_{2}^{2}\right)\left(n_{3}^{2}-n_{2}^{2}\right)}{n_{2}^{2}} \sin ^{2}\left(\frac{n_{2} \omega d}{c}\right)\right] (9.199)
T=1 \Rightarrow \sin k d=0 \Rightarrow k d=0, \pi, 2 \pi \ldots The minimum (nonzero) thickness is d=\pi / k . \text { But } k=\omega / v=2 \pi \nu / v=2 \pi \nu n / c, \text { and } n=\sqrt{\epsilon \mu / \epsilon_{0} \mu_{0}} (Eq. 9.69), where (presumably) \mu \approx \mu_{0} . \text { So } n=\sqrt{\epsilon / \epsilon_{0}}=\sqrt{\epsilon_{r}} , and hence d=\frac{\pi c}{2 \pi \nu \sqrt{\epsilon_{r}}}=\frac{c}{2 \nu \sqrt{\epsilon_{r}}}=\frac{3 \times 10^{8}}{2\left(10 \times 10^{9}\right) \sqrt{2.5}}=9.49 \times 10^{-3} m , \text { or } 9.5 mm .
n \equiv \sqrt{\frac{\epsilon \mu}{\epsilon_{0} \mu_{0}}} (9.69)