= Radially layered coaxial cable
For a <coaxial cable> with inner and outer radii $a,b$, a radial interface $c$, and <permittivities> $\epsilon,\epsilon'$, define $T=\ln(c/a)/\epsilon+\ln(b/c)/\epsilon'$. With terminal <electric potential difference> $V$, the <electric displacement field> is $D_\rho=V/(T\rho)$. The <electric field> is $E_\rho=V/(T\epsilon_i\rho)$ in each layer $i$. If the outward and return <electric currents> have magnitudes $I$, the <magnetic field strength> in the gap is $H_\varphi=I/(2\pi\rho)$ and the <magnetic field> is $B_\varphi=\mu_iI/(2\pi\rho)$.
The proof uses <Gauss's law> without free interface <electric charge>, the <electric potential difference> integral, and the <magnetostatic Ampère law>. The axial <Poynting vector> is $VI/(2\pi T\epsilon_i\rho^2)$, so its cross-sectional <surface integral> is $VI$, independently of the radial material layering and <magnetic permeabilities>. The empty inner and exterior regions have zero <electric field> and <magnetic field> for the regular solution with bounded exterior <electric potential> and no applied <magnetic field>.
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