Exercise 5.5.5: The field of a coaxial line

In cylindrical coordinates

Write $E_{\rho }=F(z,t)/\rho$ with $F=E_{0}a\cos (\omega t-kz)$. The scalar Laplacian in cylindrical coordinates gives

since $\rho d(1/\rho )/d\rho =-1/\rho$ and $\frac{1}{\rho }\frac{d}{d\rho }(-1/\rho )=1/{\rho }^3$. With ${\partial }^{2}F/\partial t^2=-{\omega }^{2}F$:

which cannot vanish for all $\rho$, whatever $k$ is.

$a_{\rho }=(x{a}_x+y{a}_y)/\rho$, so

The functions $x/(x^2+y^2)=\cos \varphi /\rho$ and $y/(x^2+y^2)=\sin \varphi /\rho$ are harmonic in the plane, ${\nabla }^2({\rho }^{-1}\cos \varphi )=(1-1){\rho }^{-3}\cos \varphi =0$. Hence ${\nabla }^2{E}_x=\frac{x}{{\rho }^2}{\partial }^{2}F/\partial z^2=-k^2{E}_x$ and

and the same for $E_y$. The vector field satisfies the vector wave equation although its cylindrical component does not satisfy the scalar one: the extra $1/{\rho }^3$ term in (b) is exactly the term $-E_{\rho }/{\rho }^2$ of the vector Laplacian, $({\nabla }^{2}E{)}_{\rho }={\nabla }^2{E}_{\rho }-E_{\rho }/{\rho }^2-(2/{\rho }^2)\partial E_{\varphi }/\partial \varphi$.

$v=c/1.5=2\times 10^{8}m/s$, $k=\omega /v=2\pi \times 10^9/(2\times 10^8)=10\pi rad/m$, $\lambda =v/f=20\mathrm{cm}$.

$k\lambda =10\pi \times 0.2=2\pi$, and $v=f\lambda =10^9\times 0.2=2\times 10^{8}m/s$.