Introduction to Electrodynamics · Problem 9.20

Electromagnetic Waves — David J. Griffiths

Physics · skin-depth-conductors

Problem

(a) Show that the skin depth in a poor conductor (

\sigma \ll \omega\epsilon

) is

(2/\sigma)\sqrt{\epsilon/\mu}

(independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of

\epsilon

\mu

, and

\sigma

; your answers will be valid, then, only at relatively low frequencies.) (b) Show that the skin depth in a good conductor (

\sigma \gg \omega\epsilon

) is

\lambda/2\pi

(where

\lambda

is the wavelength in the conductor). Find the skin depth (in nanometers) for a typical metal (

\sigma \approx 10^7\,(\Omega\,\text{m})^{-1}

) in the visible range (

\omega \approx 10^{15}

/s), assuming

\epsilon \approx \epsilon_0

and

\mu \approx \mu_0

. Why are metals opaque? (c) Show that in a good conductor the magnetic field lags the electric field by

45^\circ

, and find the ratio of their amplitudes. For a numerical example, use the ``typical metal'' in part (b).