PHAS3201 EM Theory Conducting Media (UCL)

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```					Electromagnetic Theory: PHAS3201, Winter 2008
7. Waves in Conducting Media
1    Conductors
Origins
• All effects stem from the wave equation:

2             ∂E    ∂2E
E − gµ       − µ 2 =0              (1)
∂t    ∂t
• In a conducting medium, J = gE, with conductance g
• Now see the effect on the plane wave:
E = E0 exp i (k · r − ωt)              (2)

TAKE NOTES

Dispersion Relation
• The dispersion relation is:
ig
k2 = µ ω2 1 +                      (3)
ω
• We get a variation of k (or λ) with ω
• Remember that vg = dω/dk and vp = ω/k
• g → 0: poor conductor, so k 2 = µ ω 2 and vp = vg

Good Conductors
• If g >> ω, we have a good conductor, and

k 2 = iµgω ⇒ k = + iµgω                 (4)
√
• What is       i?
• We write:
√               π   1
2                 1
i = exp i           = exp iπ/4 = √ (1 + i)   (5)
2                      2
• So we write k = kr + iki
TAKE NOTES

Skin depth
• When we put this into E, we ﬁnd:
E = E0 exp i [(kr + iki ) · r − ωt]              (6)
= E0 (exp −ki · r) (exp i [kr · r − ωt])    (7)

• This is a normal travelling wave
• It is exponentially damped in the direction of k
• We deﬁne an attenuation:
E0 (d) = E0 (0) exp (−d/δ)              (8)

PHAS3201 Winter 2008                Section VII. Waves in Conducting Media             1
PHAS3201: Electromagnetic Theory

Skin Depth

• We deﬁne the skin depth:
1         2
δ=      =                                                    (9)
ki       µωg
• An EM wave falling from air to good conductor will penetrate a few δ
• For copper δ = 8.5mm at 60Hz, δ = 7.1µm at 100 MHz

• Hence waveguides conﬁne EM waves to the space around conductors

2    Reﬂection At Metal Surface
We start with the refractive index, n, which can be (and will be here !) complex.

Refractive Index

• We know that n = ck/ω
• If we substitute in from the results above, assuming µ = µ0 we get:

c√          1+i           g
n=       µgω       √     =              (1 + i)                          (10)
ω              2         2ω   0

• A “good” conductor, as deﬁned earlier, has g >> ω
• Here, |n| >> 1
• Consider normal incidence at a metal surface

TAKE NOTES

3    Plasmas
A neutral plasma can be thought of as a group of massive, slowly moving positive ions with a cloud of free
electrons surrounding it (of density Ne electrons per unit volume) so that the whole system is neutral. The system
is homogeneous on macroscopic length scales, so that there are no large areas of positive or negative charge. If
there is a local ﬂuctuation, so that the electrons are displaced by x, there is a resulting polarisation, P = −Ne ex,
leading to a restoring force on the electrons.
In taking this approach we are expressing the local build up of charge density due to an electromagnetic wave
in the same way as we did for a dielectric: as an induced polarisation charge density. It’s presence will later be
expressed through an effective permittivity, so we will be justiﬁed in setting · D = 0 and also · E = 0 for a
linear medium. The wave equation for E that we have used before will therefore remain valid.

Plasma Frequency

• Consider a slab of plasma, width s, area A
• Displace all electrons in slab by x

• Produces charge build-up Q = Ne exA (equivalent to dipole moment density P)
Ne e2
• There are oscillations with frequency ωP =       me 0

TAKE NOTES

PHAS3201 VII. Waves in Conducting Media                                                                            2
PHAS3201: Electromagnetic Theory

Figure 1: A slab of plasma in which the electrons have been displaced by a small amount x.

Dispersion

• For an EM wave in a plasma, how does k depend on ω ?

• Collisions between electrons and ions are assumed infrequent
• For a high frequency wave, consider free electrons
• (only for a few cycles)
• Compare to a metal where ohmic collisions dissipate energy

• We ﬁnd:
2
ω2           ωp
k2 =         1−                                       (11)
c2           ω2

TAKE NOTES

Dispersion

• Consider a wave with ω > ωp

• k 2 > 0, so k is real and there is no attenuation
• Let us consider the group and phase velocities
• We ﬁnd:
vp vg = c2                                     (12)

• If ω < ωp , k 2 < 0 and we have absorption of energy and damping over some attenuation length, L

TAKE NOTES

PHAS3201 VII. Waves in Conducting Media                                                                   3

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