2B29 Electromagnetic Theory 11. Waves in Conducting Media 11.1 Getting the Dispersion Relation In section 7 above, for a linear conducting medium with J = σ E , D = ε E and no free charge concentrations, the wave equation is ∂E ∂ 2E ∇ 2 E − σµ − εµ 2 = 0 . (7.2) ∂t ∂t Substituting the usual expression for a plane wave E(r, t ) = E0 exp i ( k.r − ω t + φ ) we get (c.f. (7.8), (7.9) etc.) ∂E ∂2 E = −iω E, = −ω 2 E . ∂t ∂t 2 − k 2 + iωσµ + ω 2 µε = 0 So, cancelling E, iσ k 2 = µεω 2 1 + (11.1) or εω This expression is a dispersion relation. It gives the variation of k (and therefore of λ) with the frequency. It can be used to derive both: ∂ω - the group velocity vg = (11.2) ∂k ∇ 2 E = − k 2 E;
vp =
- and the phase velocity
ω k
.
(11.3)
N.B. If we let σ → 0 in (11.1), then k 2 → µεω 2 and v p = v g =
1
εµ
, as expected.
But if σ >> εω (11.1) we call the material “a good conductor” and the dispersion k 2 = i µσω relation simplifies to k = i µσω
so
(11.4)
The quantity i may be unfamiliar, but is easy enough from first principles. For a unimodular complex number z = exp iφ we z Im(z) φ z know that z = exp i . 2
If z = i = −1 = i Im(i ) . On the Argand plot i = exp i along the imaginary axis, so In (11.4) this gives
i = exp i
π 4
= {cos
Im k = Re k =
2B29 Waves in Conductors. Spring 2004.
π
π 2
Re(z) ,
π 1 + i sin } = (1 + i ) . 4 4 2 µσω
(11.5)
2
Section 11.
1
Because k is complex (c.f. case of frustrated total internal reflection, section 9.3 above) the wave in the medium has both a propagation factor and a damping factor: ˆ − ωt ] E(r, t ) = E0 exp i[(Re k + i Im k )k.r ˆ ) exp i (Re k k.r ˆ − ωt ) = E exp(− Im k k.r (11.6) 0
The second exponential is a normal travelling wave, but when it travels a distance d in the direction of k it will be reduced in amplitude by a factor exp(− Im k d ) . We usually characterise such damping by an attenuation length δ so d E0 (d ) = E0 (0) exp −
δ
1 2 = ≡ skin depth . (11.7) Im k µωσ An electromagnetic wave falling on a good conductor from air will only penetrate by a few skin depths due to the exponential attenuation. How deep it goes depends upon its frequency; for copper δ 8.5mm at 60 Hz; 7.1µ m at 100 MHz . When we consider waveguides for radio-frequency signals we will see that the electromagnetic disturbance is treated as being primarily in the space around the conductors, not inside the conductors.
δ=
and
11.2 Reflection at a metallic Surface
vacuum n = 1, µ = µ0 , ε = ε 0 , v p = c
metal σ >> εω n ' ≠ 1, µ µ0 , ε ≠ ε 0
E0, k
E0 '', k '' = −k
Waves penetrate a few skin depths δ. Below surface get complex k′
Take normal incidence for simplicity, then the boundary conditions on E and H are very simple, since both vectors lie in the plane of the surface.
2B29 Waves in Conductors. Spring 2004.
Section 11.
2
Continuity of tangential components requires Hout E0 + E0 '' = E0 ' (11.8) A B H 0 − H 0 '' = H 0 ' surface where the –ve sign on H 0 '' ensures that the C D reflected wave propagates in the –k direction; N = E × H also reverses. Remember that in Hin discussing the boundary conditions in section 2.7 we required the absence of free currents in the surface for tangential components of H to be conserved. Here there are free currents flowing in the metal, but they are not confined to the surface – they penetrate a few skin depths. So we can shrink the sides BC, DA of the loop integral to get a negligible contribution from free currents. If µ = µ 0 then the second line of (11.8) becomes
B0 − B0 '' = B0 ' . Going back to the links between the electric and magnetic parts that we derived in k section 7.2, B0 = E0 , so this becomes
ω
k
ω or
E0 −
E0 − E0 '' =
ck '
ω
k
ω
E0 '' =
E0 ' =
k'
ω
E0 '
c E0 ' = n ' E0 ' . vp '
(11.9)
In a good conductor, as we saw above at equation (11.5), the propagation vector k′ is complex, so the refractive index n′ must also be complex. Eliminating E′ between (11.9) and the first part of (11.8) E0 − E0 '' = n ' ( E0 + E0 '') so we get a reflection coefficient for normal incidence E '' 1 − n ' (11.10) rn = 0 = E0 1 + n ' Now, from (11.5) and (11.9) µ0σω ck ' c σ 1 n' = = ( Re k '+ i Im k ') = (1 + i ) = (1 + i ) . 2 2ωε 0 ω ω ω µ 0ε 0
But our definition of a good conductor has σ >> ε 0ω , so
σ >> 1 2ωε 0
−n ' = −1 . (11.11) +n ' For large conductivity the reflected amplitude has the same magnitude as the incident amplitude. rn
This simple theory works well for radio frequencies and infra red, but we can see with our own eyes that it is not such a good description at optical frequencies because many metals give coloured reflections.
2B29 Waves in Conductors. Spring 2004.
Section 11.
3