= Solution
For a thick good conductor in the normal <skin effect> regime, the <skin-effect surface resistance> is determined by
$$
\delta=\sqrt{\frac{2}{\mu\sigma\omega}},\qquad R_s=\frac1{\sigma\delta}=\sqrt{\frac{\mu\omega}{2\sigma}}.
$$
Identical guide geometry then gives
$$
\boxed{\frac{R_{\rm steel}}{R_{\rm gold}}=\sqrt{\frac{\mu_{\rm steel}\sigma_{\rm gold}}{\mu_{\rm gold}\sigma_{\rm steel}}}\simeq\sqrt{\frac{4.1\times10^7}{1.5\times10^6}}=5.23,}
$$
where the numerical ratio assumes both coatings are nonmagnetic, with $\mu\simeq\mu_0$. A magnetic stainless-steel grade would require its <magnetic permeability>; conductivity alone does not fix that alternative.
Using $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{H\,m^{-1}}$, the gold <skin depth> at the stated frequency is
$$
\boxed{\delta_{\rm gold}\simeq7.86\times10^{-7}\,\mathrm m=0.79\,\mathrm{\mu m}.}
$$
The <metal coating thickness relative to skin depth> sets the useful scale. Thus a layer of order \b[$1\,\mathrm{\mu m}$] begins to be useful, and a few <skin depths>, roughly \b[$2$–$3\,\mathrm{\mu m}$], substantially screen the poorer substrate. This is a thickness scale, not a specified attenuation threshold. It assumes a continuous coating thicker than the relevant microscopic surface scale.<Electromagnetism by David Tong>[Chapter 7: wave propagation in conductors and skin depth]{ref}<CODATA recommended values of the fundamental physical constants 2014 by National Institute of Standards and Technology>[Vacuum permeability in the then-current SI convention]{ref}
Back to article page
