§3. Electromagnetic Waves
§3.4. EM fields (waves) in conductors
The behaviour of EM waves in a conductor is quite different from that in a source-free
medium. The conduction current in a conductor is the cause of the difference. We shall
analyze the source terms in the Maxwell’s equations to simplify Maxwell’s equations in a
conductor. From this set of equations, we can derive a diffusion equation and investigate the
skin effects.
§3.4.1 Skin Effects in Conductors
A. Maxwell’s Equations in a Conductor
Complete Maxwell’s equations:
ρ
−
→
∇· E =
²
−
→
∂B
−
→
∇× E = −
∂t
−
→
∇· B = 0 
−
→
∂
E
−
→
∇× B = µ ~j + ²
∂t
Fact 1: Conducting current dominates over the displacement current
In a conductor, the electric field is the driving source for the conduction current. The
collision is the source of impedance. The conduction current is governed by the Ohm’s
law:
−
→
−
→
j f = σE
where σ (S/m) is the conductivity.
The AC conductivity differs from the DC conductivity. Let ν be the collision frequency
of electrons (current carrier in conductors) with ions and ω the frequency of the EM
waves in the conductor. The equation of motion for electrons is:
m
→
d−
v
−
→
→
= −e E − mν −
v
dt
→
→
Assume −
v =−
v 0 e−iωt and use ∂/∂t → −iω, we obtain
−
→
→
→
−iωm−
v = −e E − mν −
v
→
Phys 463, E & M III, C. Xiao
−
→
v =
−e
−
→
E
m(ν − iω)
1 / note11
−
→
→
Recall the current density is expressed by j = −en−
v (n is the electron number
density in the conductor),
ne2
−
→
−
→
jf=
E
m(ν − iω)
which gives the AC conductivity in conductor:
σ(ω) =
1 ne2
ν − iω m
In practical situations, ω ¿ ν ∼ 1014 (1/sec) (infrared range). So the DC conductivity
ne2
mν
σ=
can be used.
Let’s now compare the magnitude of conduction current with that of the displacement
current.
−
→ −
→
Assume E = E 0 e−iωt . Then
¯
→ ¯¯
¯−
¯ j f¯
σE
σ
¯ −
¯=
=
¯ ∂→
¯
²ωE
²ω
¯² E ¯
∂t
In copper, σ = 6 × 107 (S/m). The condition for jf ' ²
ω=
−
→
∂E
,
∂t
or
σ
²ω
' 1 leads to
σ
6 × 107
=
∼ 7 × 1019 (rad/sec)
²
8.85 × 10−12
At frequencies ω < 1012 (rad/sec) (communication wave frequency),
σ
À 1 or
²ω
¯−
¯
¯→ ¯
¯ j f¯ À
Ampere’s law in a conductor:
¯ −
¯
¯ →¯
¯ ∂E ¯
¯²
¯
¯
¯
¯ ∂t ¯
−
→
−
→
−
→
∇× B = µ j f = µσ E
Fact 2: No significant charge accumulation
Because of the good conductivity, no significant charge accumulation in a conductor is
expected (ρ ' 0).
Phys 463, E & M III, C. Xiao
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From the charge conservation and Gauss’s law
ρ
−
→
∇· E =
²
∂ρf
−
→
∇· j f = −
∂t
we obtain
∂ρf
σ
−
→
= −σ∇· E = − ρf
∂t
ε
So
σ
ρf = ρf (0)e− ε t
The free charge ρf (0) dissipates in a characteristic time τ = σε , similar to the static
case in which the charge will flow out to the edge of the conductor.
If the transient phase is excluded, ρf = 0 can be assumed in a conductor. For simplicity,
we shall consider “good” conductor case in which the displacement current can be
ignored.
Maxwell’s equations in a conductor:
−
→
∇· E
−
→
∇× E
−
→
∇· B
−
→
∇× B
= 0
−
→
∂B
= −
∂t
= 0
−
→
= µσ E
It can be shown that the EM waves in conductor are also TEM (Transverse EM ) waves
B. Diffusion Equation
As we did before for waves in source-free media, let’s apply curl operator to the 2nd
Maxwell’s equation:
 −
→
³
´
∂B 
∂
∂ ³ −
−
→
−
→
→´
∇× ∇× E = −∇× 
= − (∇× B ) = −
µσ E
∂t
∂t
∂t
LHS of the equation
So
³
³ −
−
→´
→´
−
→
−
→
∇× ∇× E = ∇ ∇· E − ∇2 E = −∇2 E
−
→
∂E
∇ E = µσ
Diffussion equation
∂t
Similarly, the magnetic field also satisfies the same diffusion equation:
−
→
∂B
→
2−
∇ B = µσ
Diffussion equation
∂t
→
2−
Phys 463, E & M III, C. Xiao
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C. Skin Depth
Suppose we have a plane wave. It comes from the −z direction and reaches a large
−
→
conductor surface at z = 0. Outside of a conductor: E = E0 e−iωt~ex at z = 0.
Assume the wave inside the conductor has the form
−
→ −
→
E = E 0 ei(kz−ωt)
where k is an unknown constant. Recall
∂
→ −iω
∂t
∇ → ik,
for the waves of the above type, we find from the diffusion equation
−
→
−
→
(ik)2 E = −iωµσ E
π
k 2 = iωµσ = ωµσe 2 i
→
r
ωµσ
√
k = ±e
ωµσ = ±(1 + i)
2
π
i
4
Choose “+” sign to allow the electric field to damp (to “propagate”) in the +z direction.
Separate the real and imaginary parts of k:
k = k+ + ik− ,
k+ = k− =
r
ωµσ
2
−
→ −
→
E = E 0 e−k− z ei(k+ z−ωt)
−
→ −
→
B = B 0 e−k− z ei(k+ z−ωt)
(1)
(2)
If the “good” conductor assumption is not valid, the displacement current should be
included in the 4th Maxwell’s equation. The solution for E and B are the same as
above with
s
1/2
r
µ
¶
εµ 
σ 2
1+
± 1
k± = ω
2
εω
The equations (1) and (2) indicate that the amplitude of E and B fields decays to 1/e
of their values at z = 0 in a distance:
δ=
1
k−
Phys 463, E & M III, C. Xiao
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where δ is called skin depth
For a good conductor (σ À εω):
δ=
s
2
(m)
ωµσ
Also, the wavelength is λ = 2π/k+ = 2πδ inqa good conductor. The wave decays
significantly within one wavelength. Since δ ∝ 1/ωσ, deep penetration occurs for
1. Low frequency
2. poor conductor
Example: skin depth at f = 60 Hz for copper.
δ=
s
2
= 8 × 10−3 m = 8 mm
−7
7
2π × 60 × 4π × 10 × 6 × 10
Phys 463, E & M III, C. Xiao
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There is no advantage to construct AC transmission lines using wires with a radius
much larger than the skin depth because the current flows mainly in the outer part of
the conductor.
For a poor conductor (σ ¿ εω):
2
δ=
σ
s
ε
(m)
µ
independent of the frequency.
Example:
For sea water, µ = µ0 = 4π × 10−7 N/A2 , ε ' 70²0 = 6 × 10−10 C2 /N·m2 , and σ ' 5
(Ω·m)−1 .
Sea water is a poor conductor for frequency
ω
σ
f=
À
= 109 Hz
2π
2πε
or λ ¿ 30 cm. The skin depth is
s
s
2 ε
2 70ε0
δ =
=
σ µ
σ µ0
√
√
2 70
2 70
=
=
' 1 cm
σZ
5 × 377
In the radioq frequency range (f ¿ 109 Hz) sea water is a good conductor, the skin
depth δ = 2/(ωµσ) is quite short. To reach a depth δ = 10 m, for communication
with submarines,
ω
1
f=
' 500 Hz
=
2π
πµσδ 2
Phys 463, E & M III, C. Xiao
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The wavelength in the air is about
λ=
c
3 × 108
=
= 600 km
f
500
The required λ/4 antenna would be gigantic.
§3.4.2 Monochromatic plane waves in a Conductor
A. Transverse waves
The E and B fields in a conductor
~ t) = B
~ 0 e−k− z ei(k+ z−ωt)
B(z,
~ t) = E
~ 0 e−k− z ei(k+ z−ωt) ,
E(z,
can be rewritten as
~ t) = E
~ 0 ei(kz−ωt) ,
E(z,
~ t) = B
~ 0 ei(kz−ωt)
B(z,
They have the same functions as EM wave in vacuum, except that k is a complex
number Following the same calculation for waves in vacuum, we can derive from
~ = 0,
∇·E
~ =0
∇·B
and
the following results:
~k · E
~ = k~ez · E
~ =0
~k · B
~ = k~ez · B
~ =0
~ and B
~ are perpendicular to ~ez , the wave propagation direction.
Both E
Let’s assume
~ = E0 ei(kz−ωt)~ex = E0 e−k− z ei(k+ z−ωt)~ex
E
From
~
~ = − ∂B
∇×E
∂t
we obtain
~ = k E0 e−k− z ei(k+ z−ωt)~ey
B
ω
|k| iφ
=
e E0 e−k− z ei(k+ z−ωt)~ey
ω
where
Ã
!
k−
φ = tan
,
k+
√
For a good conductor φ = 450 , |k| = ωµσ
−1
|k| =
q
2
2
k+
+ k−
The B fields lags behind the electric fields
Phys 463, E & M III, C. Xiao
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§4.3.3 Reflection of EM Waves on a Conductor Surface
We have seen that the EM waves do not penetrate the conductor deeply. Where do
the waves go? Absorbed or reflected?
Back to our example with a plane wave perpendicularly propagating to a conducting
surface
~ = Hy~ey = H0 ei(z/δ−ωt) e−z/δ~ey
H
~ = ~j = σEx~ex = σE0 ei(z/δ−ωt) e−z/δ~ex
∇×H
~ x = [∇ × (Hy~ey )]x = ∂Hz − ∂Hy = 1 − i Hy
(∇ × H)
∂y
∂z
δ
| {z }
=0
So
1−i
Hy = σEx
δ
Phys 463, E & M III, C. Xiao
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Characteristic impedance of the conductor
Z =
Ex
1−i
1−i
=
=
Hy
σδ
σ
1−i
= √
2
r
ωµ
=
σ
s
r
ωµσ
2
−iωµ
σ
For 4mm microwave, f = 75 Ghz. If the conductor is aluminum (σAl = 2 × 107 S/m)
r
s
ωµ
2π × 75 × 109 × 4π × 10−7
|ZAl | =
=
σ
2 × 107
= 0.17 Ω ¿ Zair = 377 Ω
The reflectivity:
Γ=
ZAl − Zair
−Zair
'
= −1
ZAl + Zair
Zair
Almost complete reflection.
Phys 463, E & M III, C. Xiao
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