A coaxial cable transmission line is known to have a characteristic impedance of 50 $$Omega$$. Measurement of the capacitance between the center conductor and the outer shield indicates a capacitance per unit length of 133 pF/m. The inductance per unit length of the coaxial cable is most nearly:

A.  6.6 nH/m
B.  33 nH/m
C.  0.33 $$mu$$H/m
D.  3.3 $$mu$$H/m

The characteristic impedance of a lossy coaxial cable is determined as follows:

$z_0=\sqrt{\frac{R\text{'} + j\omega L\text{'}}{G\text{'} + j\omega C\text{'}}}$

where 

$R\text{'} =$the resistance per unit length

$G\text{'} =$the conductance of the dielectric per unit length

$L\text{'} =$the inductance per unit length

$C\text{'} =$the capacitance per unit length

$j =$imaginary unit

$\omega =$angular frequency

In a lossless coaxial cable, in the case where $R\text{'}$ and $G\text{'}$ are not specified, we can assume that they are both equal to 0.  

Simplifying the equation we now have:

$z_0=\sqrt{\frac{j\omega L\text{'}}{j\omega C\text{'}}} = \sqrt{\frac{\cancel{j\omega }L\text{'}}{\cancel{j\omega }C\text{'}}} = \sqrt{\frac{L\text{'}}{C\text{'}}}$

Since we are solving for $L\text{'}$we can square both sides to remove the square root.

$z_0^2 = \frac{L\text{'}}{C\text{'}}$

Next, we can multiply both sides by $C\text{'}$and solve for $L\text{'}$with our known values as follows:

$$\begin{gathered} z_0^2*C\text{'} = L\text{'} = {(50 \Omega )}^2 * (133 pF/m) \\ =(2500 {\Omega }^2)(133 pF/m) \\ =3.325 * {10}^{-7} H/m \\ \approx 0.33 \mu H/m \end{gathered}$$