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