Answer 7.5

_psets/5.md

@@ -125,17 +125,43 @@ inner radius of 0.406 mm, and an outer radius of 1.48 mm.
 {:.question}
 What is the characteristic impedance?
 
+$$
+\begin{align*}
+Z &= \sqrt{\frac{L}{C}} \\
+&= \sqrt{\frac{\mu_0}{2 \pi} \ln \left( \frac{r_o}{r_i} \right)
+    \frac{1}{2 \pi \epsilon_0 \epsilon_r} \ln \left( \frac{r_o}{r_i} \right)} \\
+&= \frac{1}{2 \pi} \sqrt{\frac{\mu_0}{\epsilon_0 \epsilon_r}} \ln \left( \frac{r_o}{r_i} \right) \\
+&= 51.6 \si{\ohm}
+\end{align*}
+$$
+
 ### (b)
 
 {:.question}
 What is the velocity?
 
+$$
+\begin{align*}
+v &= \frac{1}{\sqrt{L C}} \\
+&= \frac{1}{\sqrt{\frac{\mu_0 \ln(r_o / r_i)}{2 \pi}
+    \frac{2 \pi \epsilon_0 \epsilon_r}{\ln( r_o / r_i )}}} \\
+&= \frac{1}{\sqrt{\mu_0 \epsilon_0 \epsilon}} \\
+&= \num{2e8} \si{m/s}
+\end{align*}
+$$
+
+This is two thirds the speed of light.
+
 ### (c)
 
 {:.question}
 If a computer has a clock speed of 1 ns, how long can a length of RG58/U be and still deliver a
 pulse within one clock cycle?
 
+$$
+\num{1e-9} \si{s} \cdot \num{2e8} \si{m/s} = 0.2 \si{m}
+$$
+
 ### (d)
 
 {:.question}
@@ -143,12 +169,24 @@ It is often desirable to use thinner coaxial cable to minimize size or weight bu
 impedance of RG58/U (to minimize reflections). If such a cable has an outer diameter of 30 mils (a
 mil is a thousandth of an inch), what is the inner diameter?
 
+Solving the formula above for $$r_i$$,
+
+$$
+r_i = r_0 \cdot e^{-2 \pi \cdot 51.6 \sqrt{\epsilon_0 \epsilon_r / \mu_0}}
+$$
+
 ### (e)
 
 {:.question}
 For RG58/U, at what frequency does the wavelength become comparable to
 the diameter?
 
+We know $$\lambda \nu = c$$, so
+
+$$
+\num{2e8} \si{m/s} \cdot \frac{1}{2 \cdot \num{1.48e-3} \si{m}} = \num{6.76e10} \si{Hz}
+$$
+
 
 ## (7.6)