Consider a plane wave normally incident on a half-space of copper. If $f=$ I $\mathrm{GHz}$, compute the propagation constant, impedance, and skin depth for the conductor. Also, compute the reflection and transmission coefficients.

Solution:

For copper, $\sigma=5.813 \times 10^7 \mathrm{~S} / \mathrm{m}$, so the skin depth is

$$ \delta_s=\sqrt{\frac{2}{u^\mu \mu}}=2.088 \times 10^{-6} \mathrm{~m} $$

and the propagation constant is, , $$ \gamma=\frac{1+j}{\delta_s}=(4.789+j 4.789) \times 10^5 \mathrm{~m}^{-1} $$

The intrinsic impedance is, From,

$$ \eta=\frac{1+j}{\sigma \delta_3}=(8.239+j 8.239) \times 10^{-3} \Omega $$

which is quite small relative to the impedance of free-space $(\eta)=377 \Omega)$. The reflection coefficient is then,

$$ \mathrm{T}=\frac{\eta-\eta_0}{\eta+\eta_n}=1 \angle 179.99^{\circ} $$

(practically that of an ideal short circuit), and the transmission coefficient is,

$$ T=\frac{2 \eta}{\eta+\eta_0}=6.181 \times 10^{-5} \angle 45^5 $$