(Solved): (IV.2) Coaxial cable (20 points) (a) For a coaxial transmission line with the cross section shown i ...

(IV.2) Coaxial cable (20 points) (a) For a coaxial transmission line with the cross section shown in the figure. Assume that the dielectric medium filled in between the conductors has a dielectric constant $?$ and permeability $?$. Show that the capacitance-per-unit-length $C_l$ and the inductance-per-unit-length $L_l$ have the following forms: $C_l=\frac{?}{2\ln (b/a)}{L}_l=\frac{2?}{c^2}\ln \frac{b}{a}$ By definition: $C=\frac{Q}{V}$ and $L=\frac{1}{c}?$ in the Gaussian units. (b) Find the speed of propagation and the characteristic impedance. (c) Coaxial cables used for high-frequency signals (such as cable TV) often consist of a thin copper wire in a polyethylene sleeve on which a flexible copper braid is woven (with a protective plastic jacket over all). A popular example (trade-named $RG?58$ ) has a centerconductor diameter of $0.085\mathrm{cm}$, dielectric constant $?=2.3(?=1)$, and characteristic impedance $Z_0=50?$. Find the diameter of the copper braid and the speed of propagation expressed as a percentage of the speed of light in vacuum. (d) For a transmitting voltage of the form $V(z,t)=V_0{e}^{i(kz??t)}$, obtain the expressions for current $I(z,t)$, electric field $E(r,z,t)$ and $B(r,z,t)$. (e) Calculate the total power flow by integrating the average Poynting vector, and verify that the power flow is $V_0{I}_0^{?}/2$, the result from simple circuit theory. This shows that the flow of power in a transmission line takes place entirely via the electric and magnetic fields between the two conductors; power is not transmitted through the conductors themselves.