Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 2A: AC Resistance of a Round Wire When the Skin Depth δ Is Comparable to the Radius a of the Wire

We have shown in Section 2.2 that the parameters α and β are given by δ⁻¹ in a good conductor. The approximations made in arriving at this result assumed that the loss tangent T = σ_c/ωε is large. Another way of stating the approximation is to say that the displacement current density $\text{j}\mathrm{\omega }\mathrm{\epsilon }\tilde{E}$ is neglected in comparison with the conduction current density ${\mathrm{\sigma }}_c\tilde{E}$. The propagation constant γ = jk may be obtained in this case by neglecting the first term on the RHS of (2.4)

(2A.1)$${\mathrm{\gamma }}^2=-k^2=\text{j}\mathrm{\omega }\mathrm{\mu }\mathrm{\sigma }=2\mathrm{\pi }f\mathrm{\mu }\mathrm{\sigma }{\text{e}}^{j\mathrm{\pi }/2}.$$

The wave equation for the current density $\tilde{J}={\mathrm{\sigma }}_{\text{c}}\tilde{E}$ is given by

(2A.2)$${\nabla }^2\tilde{J}-\text{j}\mathrm{\omega }\mathrm{\mu }{\mathrm{\sigma }}_{\text{c}}\tilde{J}=0.$$

For the cylindrical one-dimensional problem under consideration ...