E&M 2 (16/1) W.-Y. Choi Lect. 8: Plane Waves in Lossy Media H B E D 0 D E j B B H J j D Maxwell’s Equations We assumed for EM wave equation derivation 0 J What if E-field generates currents? , 0 J E (Cheng, 8-3) Lossy medium
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
HB
ED
0
D
E j B
B
H J j D
Maxwell’s Equations
We assumed
for EM wave equation derivation
0J
What if E-field generates currents?
, 0J E
(Cheng, 8-3)
Lossy medium
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
HB
ED
Maxwell’s Equations
, 0J E
H σE jω E
cσε ε - j
( σjω j )Eω
cjω E
Complex !
0
D
E j B
B
H J j D
'' ''' ,cε j
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
Wave number for c=’-j’’ ?
ck μ : attenuation constant
: propagation constant
What does complex k mean?
0jkzE xE e Exponential Decay!
12
' 1'
με j
,β jα
0
j j zxE e 0j z zxE e e
Consider plane wave solution:
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
zzjzjjjkz eeExeExeExE 000Exponential Decay!
- How does affect wave propagation?
- How much does the wave propagate into the medium?
Penetration depth,Or Skin depth
- What is wavelength in lossy medium?
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
'''cσε ε - j j
1 : penetration depth,
Complex permittivity:
- How much loss does the medium have?
In medium where currents can flow (non-zero )
Complex wave number: ,k β jα
Wave experiences loss: Lossy Medium
''
ctan :'
loss angle, loss tangent
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
2'' ''1 1' 1 ........2 ' 8 '
k j
12 ''' 1 ,
'ck μ με j β jα
If <<1 ’’<<1
1/2 21 1Using (1 ) 12 8
x x x
Complicated Expression
Use approximation:
2''1' 18 '
''
'2 '
12 '
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
2
''
1 "' 18 '
1'' 2 '
Phase velocity? pv
c
Impedance?
Complex impedance?
12
1' '
j
''
1' 2 '
j
12 ''' 1 ,
'k με j β jα
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
When σ≫1
ck μ
f
'' 1
' 2' j
12
j
'' 112
j
1f j j
''
1'
1
'' 2' 1 ,
'με j
''( )
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
When σ≫1
- increases with frequency
; ck μ j f
increases with frequency
Higher frequency signals have larger loss in good conductors
Higher frequency signals have smaller wavelength in good conductors
- Penetration depth?
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
When σ≫1,
c
E and H have phase difference of 45°
121
j
''j
1 1
2j
Impedance?
1f j
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
MaterialConductivity(S/m)
ƒ = 60(Hz) 1(MHz) 1(GHz)
Silver 6.17 x 107 8.27(mm) 0.064(mm) 0.0020(mm)
Copper 5.80 x 107 8.53 0.066 0.0021
Gold 4.10 x 107 10.14 0.079 0.0025Aluminum 3.54 x 107 10.92 0.084 0.0027Iron
(μr ≅ 103)1.00 x 107 0.65 0.005 0.0016
Seawater 4 32(m) 0.25(m)
Penetration depths for various materials
E&M 2 (16/1) W.-Y. Choi
Lect. 8: Plane Waves in Lossy Media
Question: How do submarines inside deep sea communicate?
-Extremely Low Frequency Communication: 76, 82Hz
-Transmitter Antenna size: several tens of km!
- Not much bandwidth
Wavelength: ~ 3,600 km