(a) (b)Figure 71: a) Dielectric constantεvs.ω. The dielectric constant is positive for frequencies above the
plasma frequency; b) Reflectivity of Aluminium, at about 15 eV it drops, this is where the
frequency of the light is the plasma frequency.
10.5 Skin depth
The skin depth is an important property in microwave engineering and is related to the high frequency
properties of metal. In this case high frequency means that the frequency is still below the plasma
frequency so waves do not propagate in the material. Withωthe frequency andτthe scattering time
we can say that belowωτ 1 electromagnetic waves penetrate into a material to a certain depth
which is called the skin depth.
With the curl of Ohm’s law (J = σE), Faraday’s law (∇×E = −ddtB), Ampere’s law (∇×B= μ 0 J)
and the vector identity∇×∇×B = ∇(∇·B)
︸ ︷︷ ︸
=0
−∇^2 Bthe following equation can be obtained:1
σμ 0= ∇^2 B =
dB
dt(121)
Once again we assume harmonic solutionsB 0 = exp [i(kx−ωt)], with this the equation forkcan
be calculated to be:
k =√
iωσμ 0 =√
ωσμ 0
2+i√
ωσμ 0
︸ ︷︷^2 ︸
exp.decay(122)
The inverse (units) of the component which is describing the exponential decay is a length, the skin
depth
δ =√
2
μ 0 σω. (123)
In microwave engineering (GHz) the surface resistance of a transmission line is an important property,
it’s dependent on the skin depth. The resistance of a transmission line can be written as
R =
l
σwt