1. Surface Impedance of Metals
Section 87

5. E = -(c/we) k x H where k = Ö(em)w/c

6. Inside and near a metal surface

7. Since Et and Ht are continuous at the surface, they are related by the same expression just outside

8. Surface impedance
At low w

9. Time averaged energy flux through the metal surface

11. As frequency increases, penetration depth eventually becomes comparable to the electron mean free path.
The field is then too non-uniform to use the macroscopic description based on e.
Fields don’t satisfy the macroscopic Maxwell equations in the metal
This is the only possible linear relation between the axial vector H and the polar vector E.

13. As frequency increases further…

14. At high frequency, the distance traveled by conduction electrons during one electromagnetic wave period
Then we can neglect the spatial inhomogeneity of the field relative to the electron motion

16. and if m is real (no magnetic dispersion)
holds
and if m is real (no magnetic dispersion)
while e is complex (significant dispersion)
Then the condition e” > 0 requires z’z” < 0,
And since z’ >0, we must have z” < 0.
On the other hand, if m is complex and e is real, then z” > 0.

17. Proof of first inequality

18. For superconductors, the penetration depth d is very small even for DC fields w = 0.
For small w, the distribution of the H-field is the same as in the static case.
(59.2)
Defines d
Boundary condition
For a superconductor. Pure Imaginary. No loss.

19. Surface impedance has dispersion.
Consider complex z to be a function of a complex frequency w.
Surface impedance integral operator
Et depends on values of Ht at previous times.
This means that z(w) must be regular in the upper half plane of complex w.
z(w) is regular on the real w axis except at w = 0.
If Ht is real, then et must be real: z(-w*) = z*(w)

20. Energy dissipation is determined by z’
These properties give Kramers-Kronig relations between z’ and z”

21. A use of surface impedance is to calculate reflection from metals.
For perpendicular polarization

22. Boundary condition
superposition

23. is small for metals

25. Parallel polarization