Electromagnetic wave propagations in different media

1. Course: Electromagnetic Theory
paper code: EI 503
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
Topic: EM Wave Propagation
18-12-2021 Arpan Deyasi, EM Theory 1
Arpan Deyasi
Electromagnetic
Theory

2. 18-12-2021 Arpan Deyasi, EM Theory 2
Maxwell’s equation in differential form
. ....(1)
E


 =
. 0 ....(2)
B
 =
....(3)
B
E
t

 = −

....(4)
C
E
B J
t
 

 = +

Arpan Deyasi
Electromagnetic
Theory

3. Propagation of wave in
[i] free space
[ii] dielectric medium
[iii] conducting medium
18-12-2021 Arpan Deyasi, EM Theory 3
Arpan Deyasi
Electromagnetic
Theory

4. 18-12-2021 Arpan Deyasi, EM Theory 4
Propagation of Electromagnetic
Wave at Free Space
Arpan Deyasi
Electromagnetic
Theory

5. Maxwell’s equation in free space
0 & 0
J
 = =
In free space
Maxwell equation becomes
. 0 ....(1)
E
 = . 0 ....(3)
B
 =
....(2)
B
E
t

 = −
 0 0 ....(4)
E
B
t
 

 =

18-12-2021 Arpan Deyasi, EM Theory 5
Arpan Deyasi
Electromagnetic
Theory

6. From third equation
B
E
t

 = −

( ) B
E
t

  = −

( ) ( )
2
.E E B
t

  − = − 

( ) ( )
2
.E E B
t

  − = − 

Maxwell’s equation in free space
18-12-2021 Arpan Deyasi, EM Theory 6
Arpan Deyasi
Electromagnetic
Theory

7. 2
0 0
E
E
t t
 
 
 
 =  
 
 
Maxwell’s equation in free space
2
2
0 0 2
0
E
E
t
 

 − =

Equation for electric field in free space
18-12-2021 Arpan Deyasi, EM Theory 7
Arpan Deyasi
Electromagnetic
Theory

8. From fourth equation
0 0
E
B
t
 

 =

( ) 0 0
E
B
t
 
 

  =  

 
( ) 2
0 0
.
E
B B
t
 
 

  − = 
 

 
Maxwell’s equation in free space
( )
2
0 0
B E
t
 

 = − 

18-12-2021 Arpan Deyasi, EM Theory 8
Arpan Deyasi
Electromagnetic
Theory

9. Maxwell’s equation in free space
2
0 0 .
B
B
t t
 
 
 =
 
2
2
0 0 2
0
B
B
t
 

 − =

Equation for magnetic field in free space
18-12-2021 Arpan Deyasi, EM Theory 9
Arpan Deyasi
Electromagnetic
Theory

10. Generalized equation
2
2
2 2
1
0
U
U
c t

 − =

Maxwell’s equation in free space
18-12-2021 Arpan Deyasi, EM Theory 10
Arpan Deyasi
Electromagnetic
Theory

11. Relation between field and propagation vectors
Solution of wave equations are
( )
0
( , ) exp .
E r t E i k r t

 
= −
 
( )
0
( , ) exp .
B r t B i k r t

 
= −
 
( )
0
. . exp .
E E i k r t

 
 = −
 
Maxwell’s equation in free space: Relative directions
Taking divergence of both the sides
18-12-2021 Arpan Deyasi, EM Theory 11
Arpan Deyasi
Electromagnetic
Theory

12. ( )
. . exp .
E ik E i k r t

 
 = −
 
Maxwell’s equation in free space: Relative directions
In free space
. 0
E
 =
. 0
k E =
Similarly
( )
. . exp .
B ik B i k r t

 
 = −
 
18-12-2021 Arpan Deyasi, EM Theory 12
Arpan Deyasi
Electromagnetic
Theory

13. E k
⊥
Maxwell’s equation in free space: Relative directions
. 0
B
 =
. 0
k B =
B k
⊥
Similarly
18-12-2021 Arpan Deyasi, EM Theory 13
Arpan Deyasi
Electromagnetic
Theory

14. Maxwell’s equation in free space: Relative directions
Again
0 0
E
B
t
 

 =

( )
( ) ( )
( )
0 0 0 0
exp . exp .
B i k r t E i k r t
t
   

   
 − = −
   

0 0
1
E k B
  
= − 
E B
⊥
All field components and propagation vector are mutually orthogonal
18-12-2021 Arpan Deyasi, EM Theory 14
Arpan Deyasi
Electromagnetic
Theory

15. Maxwell’s equation in free space: Poynting’s vector
Poynting vector in free space
( )
0
1
S E B

= 
k E
B


=
S E H
= 
Now
18-12-2021 Arpan Deyasi, EM Theory 15
Arpan Deyasi
Electromagnetic
Theory

16. ( )
0
1
ˆ
S E n E
c
 
 =  
 
2
0
ˆ
E
S n
c
=
2 0
0
ˆ
S E n


=
Maxwell’s equation in free space: Poynting’s vector
18-12-2021 Arpan Deyasi, EM Theory 16
Arpan Deyasi
Electromagnetic
Theory

17. 18-12-2021 Arpan Deyasi, EM Theory 17
Problem 1
Calculate Poynting vector in free space when electric field is 10 V/m
Soln
2 0
0
S E


=
12
7
8.854 10
10 10
4 10
S

−
−

=  

2
0.265 /
S W m
=
Arpan Deyasi
Electromagnetic
Theory

18. Ratio of electrostatic and magnetostatic energy is given by
2
0
2
0
1
2
1
2
e
m
E
U
U B


=
2
0 0 2
1
e
m
U E
U B
 
= =
e m
U U
=
Maxwell’s equation in free space: Energy
18-12-2021 Arpan Deyasi, EM Theory 18
Arpan Deyasi
Electromagnetic
Theory

19. k E
B


=
k E
B
kc

=
( )
0 0
ˆ
B n E
 
= 
Wave impedance 0
0
0 0 0
1
E
E
Z
B
B  
= = =
Maxwell’s equation in free space: Wave impedance
18-12-2021 Arpan Deyasi, EM Theory 19
Arpan Deyasi
Electromagnetic
Theory

20. Maxwell’s equation in free space: Propagation characteristics
2
2
0 0 2
0
E
E
t
 

 − =

2
2
0 0 2
0
B
B
t
 

 − =

Wave equations are
Solution of wave equations are
( )
0
( , ) exp .
E r t E i k r t

 
= −
 
( )
0
( , ) exp .
B r t B i k r t

 
= −
 
18-12-2021 Arpan Deyasi, EM Theory 20
Arpan Deyasi
Electromagnetic
Theory

21. Maxwell’s equation in free space: Propagation characteristics
Substituting
2 2 2
0 0
E E E
   
 = − =
2 2 2
0 0
B B B
   
 = − =
Propagation constant
2
0 0 0 0
j j
       
= − = =
18-12-2021 Arpan Deyasi, EM Theory 21
Arpan Deyasi
Electromagnetic
Theory

22. Maxwell’s equation in free space: Propagation characteristics
p
v c


= =
Phase velocity
Wavelength
0 0
2 2
 

   
= =
18-12-2021 Arpan Deyasi, EM Theory 22
Arpan Deyasi
Electromagnetic
Theory

23. 18-12-2021 Arpan Deyasi, EM Theory 23
Problem 2
Electric field in free space is
8
100cos(10 ) /
E t z V m

= +
Calculate [i] phase constant, [ii] time required to travel λ/4
Soln
In free space
8
3 10 /sec
p
v c m
= = 
8
8
10
3 10
p
v

 = =

0.33 /
rad m
 =
Arpan Deyasi
Electromagnetic
Theory

24. 18-12-2021 Arpan Deyasi, EM Theory 24
To travel a distance λ/4, it will take ¼ of its time period
2
4 4 2
T
t
 
 
= = =
8
10
2
t
 −
= 
15.71 s
t n
=
Arpan Deyasi
Electromagnetic
Theory

25. Maxwell’s equation in free space: Characteristic impedance
B
E
t

 = −

ˆ
ˆ y
z
E
E B
j k
x x t

 
− + = −
  
ˆ ˆ
ˆ ˆ ....(1)
y y
z z
E B
E B
j k j k
x x t t
 
 
− + = − −
   
From Maxwell’s equation
18-12-2021 Arpan Deyasi, EM Theory 25
Arpan Deyasi
Electromagnetic
Theory

26. Maxwell’s equation in free space: Characteristic impedance
0 0
E
B
t
 

 =

0 0
ˆ
ˆ y
z
B
B E
j k
x x t
 

 
− + =
  
0 0 0 0
ˆ ˆ
ˆ ˆ ....(2)
y y
z z
B E
B E
j k j k
x x t t
   
 
 
− + = +
   
Again from Maxwell’s equation
18-12-2021 Arpan Deyasi, EM Theory 26
Arpan Deyasi
Electromagnetic
Theory

27. Maxwell’s equation in free space: Characteristic impedance
0 0
y
z
E
B
x t
 


− =
 
0 0
y z
B E
x t
 
 
=
 
y
z
B
E
x t


=
 
y z
E B
x t
 
= −
 
From Eq. (1)
From Eq. (2)
18-12-2021 Arpan Deyasi, EM Theory 27
Arpan Deyasi
Electromagnetic
Theory

28. Maxwell’s equation in free space: Characteristic impedance
Let
( )
y
E f x vt
= −
'
y
E
vf
t

= −

0 0 '
z
B
vf
x
 

 =

0 0 '
z
B vf dx
 
= 
18-12-2021 Arpan Deyasi, EM Theory 28
Arpan Deyasi
Electromagnetic
Theory

29. 0 0
0 0
1
z
f
H dx
x
 
 

=


Maxwell’s equation in free space: Characteristic impedance
0
0
z
H f


=
0
0
z y
H E


=
18-12-2021 Arpan Deyasi, EM Theory 29
Arpan Deyasi
Electromagnetic
Theory

30. 0
0
120
y
z
E
H



= =
Maxwell’s equation in free space: Characteristic impedance
120
 
= 
18-12-2021 Arpan Deyasi, EM Theory 30
Arpan Deyasi
Electromagnetic
Theory

31. 18-12-2021 Arpan Deyasi, EM Theory 31
Problem 3
If the magnetic field strength of a plane wave is 1 A/m, what will be the strength of
electric field in free space?
Soln
120
E
H

=
120 .
E H

=
377.4 /
E V m
=
Arpan Deyasi
Electromagnetic
Theory

32. 18-12-2021 Arpan Deyasi, EM Theory 32
Problem 4
A plane wave at free space carries a power density of 1 MW/m2. Find the magnitude of
electric and magnetic field vectors
Soln
2 0
0
S E


=
2
0
E
S
Z
=
Arpan Deyasi
Electromagnetic
Theory

33. 18-12-2021 Arpan Deyasi, EM Theory 33
2
0
E SZ
=
6
0 10 377
E SZ
= = 
19.4 /
E KV m
=
Arpan Deyasi
Electromagnetic
Theory

34. 18-12-2021 Arpan Deyasi, EM Theory 34
Magnetic field
0
E
H
Z
=
120
E
H

=
51.5 /
H A m
=
Arpan Deyasi
Electromagnetic
Theory

35. 18-12-2021 Arpan Deyasi, EM Theory 35
Propagation of Electromagnetic
Wave at Dielectric Medium
Arpan Deyasi
Electromagnetic
Theory

36. Maxwell’s equation in dielectric medium
0 & 0
J
 = =
In dielectric medium
Maxwell equation becomes
. 0 ....(1)
E
 = . 0 ....(3)
B
 =
....(2)
B
E
t

 = −

....(4)
E
B
t


 =

18-12-2021 Arpan Deyasi, EM Theory 36
Arpan Deyasi
Electromagnetic
Theory

37. From third equation
B
E
t

 = −

( ) B
E
t

  = −

( ) ( )
2
.E E B
t

  − = − 

( ) ( )
2
.E E B
t

  − = − 

Maxwell’s equation in dielectric medium
18-12-2021 Arpan Deyasi, EM Theory 37
Arpan Deyasi
Electromagnetic
Theory

38. 2 E
E
t t

 
 
 =  
 
 
Maxwell’s equation in dielectric medium
2
2
2
0
E
E
t


 − =

Equation for electric field in free space
18-12-2021 Arpan Deyasi, EM Theory 38
Arpan Deyasi
Electromagnetic
Theory

39. From fourth equation
E
B
t


 =

( ) E
B
t

 

  =  

 
( ) 2
.
E
B B
t

 

  − = 
 

 
Maxwell’s equation in dielectric medium
( )
2
B E
t


 = − 

18-12-2021 Arpan Deyasi, EM Theory 39
Arpan Deyasi
Electromagnetic
Theory

40. Maxwell’s equation in dielectric medium
2
.
B
B
t t

 
 =
 
2
2
2
0
B
B
t


 − =

Equation for magnetic field in free space
18-12-2021 Arpan Deyasi, EM Theory 40
Arpan Deyasi
Electromagnetic
Theory

41. Generalized equation
2
2
2 2
1
0
U
U
v t

 − =

Maxwell’s equation in dielectric medium
18-12-2021 Arpan Deyasi, EM Theory 41
Arpan Deyasi
Electromagnetic
Theory

42. Maxwell’s equation in dielectric medium: Poynting’s vector
Poynting vector in dielectric medium
( )
1
S E B

= 
k E
B


=
S E H
= 
Now
18-12-2021 Arpan Deyasi, EM Theory 42
Arpan Deyasi
Electromagnetic
Theory

43. ( )
1
ˆ
S E n E
c
 
 =  
 
2
ˆ
E
S n
c
=
2
ˆ
S E n


=
Maxwell’s equation in dielectric medium: Poynting’s vector
18-12-2021 Arpan Deyasi, EM Theory 43
Arpan Deyasi
Electromagnetic
Theory

44. Ratio of electrostatic and magnetostatic energy is given by
2
2
1
2
1
2
e
m
E
U
U B


=
2
2
1
e
m
U E
U B

= =
e m
U U
=
Maxwell’s equation in dielectric medium: Energy
18-12-2021 Arpan Deyasi, EM Theory 44
Arpan Deyasi
Electromagnetic
Theory

45. 18-12-2021 Arpan Deyasi, EM Theory 45
Problem 5
A plane wave travelling in a medium of εr=1 & μr=1 has an electric field of 100√π V/m.
Determine total energy density.
Soln
2
1
2
e
U E

=
2
0
1
2
e r
U E
 
=
3
139 /
e
U nJ m
=
Arpan Deyasi
Electromagnetic
Theory

46. 18-12-2021 Arpan Deyasi, EM Theory 46
2
T e m e
U U U U
= + =
3
278 /
T
U nJ m
=
Arpan Deyasi
Electromagnetic
Theory

47. k E
B


=
k E
B
kc

=
( )
ˆ
B n E

= 
Wave impedance
1
E
Z
B 
= =
Maxwell’s equation in dielectric medium: Wave impedance
18-12-2021 Arpan Deyasi, EM Theory 47
Arpan Deyasi
Electromagnetic
Theory

48. 18-12-2021 Arpan Deyasi, EM Theory 48
Problem 6
Magnetic field of a plane wave has magnitude 5 mA/m in a medium defined by εr = 4, μr = 1.
Determine average power flow and maximum energy density.
Soln
1
E
B 
=
0
0
r
r
E
H
 

  
= =
Arpan Deyasi
Electromagnetic
Theory

49. 18-12-2021 Arpan Deyasi, EM Theory 49
188.4
E
H
=
3
188.4 5 10
E −
=  
94.2 /
E mV m
=
Average power flow
2
2
avg
E
P
Z
=
Arpan Deyasi
Electromagnetic
Theory

50. 18-12-2021 Arpan Deyasi, EM Theory 50
2
2.35 /
avg
P mW m
=
Maximum energy density
2
1
2
2
E
W E

= 
2
0
E r
W E
 
=
3
31.42 /
E
W pJ m
=
Arpan Deyasi
Electromagnetic
Theory

51. Maxwell’s equation in dielectric medium: Wave impedance
0 0
1 1
r r
Z
    
= =
0 0
1 1
r r
Z
   
=
0
1
r r
Z Z
 
=
18-12-2021 Arpan Deyasi, EM Theory 51
Arpan Deyasi
Electromagnetic
Theory

52. Maxwell’s equation in dielectric medium: Propagation characteristics
p
v v


= =
Phase velocity
Wavelength
2 2
 

  
= =
18-12-2021 Arpan Deyasi, EM Theory 52
Arpan Deyasi
Electromagnetic
Theory

53. 18-12-2021 Arpan Deyasi, EM Theory 53
Problem 7
Electric field of a plane wave in a perfect dielectric medium is given by
Soln
7
12cos(2 10 0.1 ) /
E t z V m
 
=  −
Calculate [i] velocity of propagation, [ii] intrinsic impedance
7
2 10
0.1
p
v
 
 

= =
8
2 10 /sec
p
v m
= 
Arpan Deyasi
Electromagnetic
Theory

54. 18-12-2021 Arpan Deyasi, EM Theory 54
0
1
r r
Z Z
 
=
1
p
r r
v c
 
=
3
2
r
 =
As μr = 1 for perfect dielectric medium
Arpan Deyasi
Electromagnetic
Theory

55. 18-12-2021 Arpan Deyasi, EM Theory 55
0
0
r
r
 


  
= =
1
377
r


= 
251.33
 = 
Arpan Deyasi
Electromagnetic
Theory

56. 18-12-2021 Arpan Deyasi, EM Theory 56
Propagation of Electromagnetic
Wave at Conducting Medium
Arpan Deyasi
Electromagnetic
Theory

57. 18-12-2021 Arpan Deyasi, EM Theory 57
Maxwell’s equation in conducting medium
In conducting medium
Maxwell equation becomes
. 0 ....(1)
E
 = . 0 ....(3)
B
 =
....(2)
B
E
t

 = −

0 & J E
 
= =
....(4)
E
B E
t
 

 = +

Arpan Deyasi
Electromagnetic
Theory

58. 18-12-2021 Arpan Deyasi, EM Theory 58
Maxwell’s equation in conducting medium
From third equation
B
E
t

 = −

( ) B
E
t

  = −

( ) ( )
2
.E E B
t

  − = − 

( ) ( )
2
.E E B
t

  − = − 

Arpan Deyasi
Electromagnetic
Theory

59. 18-12-2021 Arpan Deyasi, EM Theory 59
Maxwell’s equation in conducting medium
2 E
E E
t t
 
 
 
 = +
 
 
 
2
2
2
0
E E
E
t t
 
 
 − − =
 
Equation for electric field in conducting medium
Arpan Deyasi
Electromagnetic
Theory

60. 18-12-2021 Arpan Deyasi, EM Theory 60
Maxwell’s equation in conducting medium
From fourth equation
( ) E
B E
t
 
 

  =  +
 

 
( ) ( )
2
.
E
B B E
t
 
 

  − =  + 
 

 
E
B E
t
 

 = +

( ) ( ) ( )
2
.B B E E
t
 

  − =  + 

Arpan Deyasi
Electromagnetic
Theory

61. 18-12-2021 Arpan Deyasi, EM Theory 61
Maxwell’s equation in conducting medium
( ) 2
.
B B
B B
t t t
 
   
  
  − = − + −
   
  
   
2
2
2
0
B B
B
t t
 
 
 − − =
 
Equation for magnetic field in conducting medium
Arpan Deyasi
Electromagnetic
Theory

62. 18-12-2021 Arpan Deyasi, EM Theory 62
Maxwell’s equation in conducting medium: characteristic equation
Solution of wave equations are
( )
0
( , ) exp .
E r t E i k r t

 
= −
 
( )
0
( , ) exp .
B r t B i k r t

 
= −
 
Substituting
2 2
0
k i  
− + − =
2 2
k i  
= −
Arpan Deyasi
Electromagnetic
Theory

63. 18-12-2021 Arpan Deyasi, EM Theory 63
Maxwell’s equation in conducting medium: characteristic equation
k i
 
= +
Solving
Let
1/2
2
1 1
1 1
2 2

  

 
 
 
= + −
 
 
 
 
 
 
1/2
2
1 1
1 1
2 2

  

 
 
 
= + +
 
 
 
 
 
 
Arpan Deyasi
Electromagnetic
Theory

64. 18-12-2021 Arpan Deyasi, EM Theory 64
Maxwell’s equation in conducting medium
CASE-I: poor conductor 1


1/2
2
1 1
1 1
2 2

  

 
 
 
 = + −
 
 
 
 
 
 
1
2

 

=
Attenuation constant is independent of frequency
Arpan Deyasi
Electromagnetic
Theory

65. 18-12-2021 Arpan Deyasi, EM Theory 65
Maxwell’s equation in conducting medium
1/2
2
1 1
1 1
2 2

  

 
 
 
= + +
 
 
 
 
 
 
2
1
1
8

  

 
 
= +
 
 
 
 
  

Conductivity has negligible effect on phase constant
CASE-I: poor conductor 1


Arpan Deyasi
Electromagnetic
Theory

66. 18-12-2021 Arpan Deyasi, EM Theory 66
Maxwell’s equation in conducting medium
CASE-II: good conductor 1


1
2

   

 
 = =  
 
2

 
= =
Both attenuation constant and phase constant are independent on material permittivity
Arpan Deyasi
Electromagnetic
Theory

67. 18-12-2021 Arpan Deyasi, EM Theory 67
Maxwell’s equation in conducting medium: Poynting’s vector
( )
1
S E B

= 
( )
1/4
2
1
ˆ
1 exp tan
2
i
B n E
 

 
−
   
   
= + 
 
   
 
   
   
1/4
2
1 2
ˆ
1 exp tan
2
i
S E n
  
  
−
   
   
 = +
 
   
 
   
   
Arpan Deyasi
Electromagnetic
Theory

68. 18-12-2021 Arpan Deyasi, EM Theory 68
Maxwell’s equation in conducting medium: Energy density
Electrostatic energy density
*
1 1
Re .
2 2
e
U E D
 
=  
 
*
1 1
Re .
4 2
e
U E E

 
=  
 
 
2
1
ˆ
exp 2 .
2
e rms
U E n r
 
= −
Arpan Deyasi
Electromagnetic
Theory

69. 18-12-2021 Arpan Deyasi, EM Theory 69
Maxwell’s equation in conducting medium: Energy density
Magnetostatic energy density
*
1 1
Re .
2 2
m
U H B
 
=  
 
*
1 1
Re .
4 2
m
U H H

 
=  
 
 
1/2
2
2
1
ˆ
1 exp 2 .
2
m rms
U E n r

 

 
 
= + −
 
 
 
 
Arpan Deyasi
Electromagnetic
Theory

70. 18-12-2021 Arpan Deyasi, EM Theory 70
Maxwell’s equation in conducting medium: Energy density
Electromagnetic energy density
e m
U U U
= +
 
1/2
2
2
1
ˆ
1 1 exp 2 .
2
rms
U E n r

 

 
 
 
 
= + + −
 
 
 
 
 
 
Electromagnetic energy density is damped during propagation in conducting medium
Arpan Deyasi
Electromagnetic
Theory

71. 18-12-2021 Arpan Deyasi, EM Theory 71
Maxwell’s equation in conducting medium: Intrinsic impedance
1/2
i
i


 
 
=  
+
 
1/2 1/2
1
1
i





   
=  
 
  +
 
 
1/2
1/4
2
1





 
 
 
 =
 
 
+
 
 
 
 
 
&
1
1
tan
2



−  
=  
 
Arpan Deyasi
Electromagnetic
Theory

72. 18-12-2021 Arpan Deyasi, EM Theory 72
Maxwell’s equation in conducting medium: Intrinsic impedance
CASE-I: poor conductor 1


1/ 2
1
1
i





  
=  
 
  +
 
 
1/2
1
2
i
 

 
   
= −
 
 
   
Arpan Deyasi
Electromagnetic
Theory

73. 18-12-2021 Arpan Deyasi, EM Theory 73
Maxwell’s equation in conducting medium: Intrinsic impedance
CASE-II: good conductor 1


1/ 2
1
1
i





  
=  
 
  +
 
 
i


=
Arpan Deyasi
Electromagnetic
Theory

74. 18-12-2021 Arpan Deyasi, EM Theory 74
Problem 8
A plane wave with frequency 2 MHz is incident upon Cu conductor having electric
field 2 mV/m. Given σ = 5.8⨯10-7 mho/m, εr = 4, μr = 1. Calculate characteristic
impedance and average power density.
Soln
7
20.85 10 1


= 



 =
4
5.235 10
 −
=  
Arpan Deyasi
Electromagnetic
Theory

75. 18-12-2021 Arpan Deyasi, EM Theory 75
Average power density
2
1
2
avg
E
P

=
2
3.82 /
avg
P mW m
=
Arpan Deyasi
Electromagnetic
Theory

76. 18-12-2021 Arpan Deyasi, EM Theory 76
Maxwell’s equation in conducting medium: Wave impedance
( )
1
B
i
E
 

= +
1/4
2
1
1 exp tan
2
B i
E
 

 
−
   
   
= +
 
   
 
   
   
1/4
2
1
1 exp tan
2
E i
Z
B
 

 
−
   
   
= = + −
 
   
 
   
   
Arpan Deyasi
Electromagnetic
Theory

77. 18-12-2021 Arpan Deyasi, EM Theory 77
Maxwell’s equation in conducting medium: Phase velocity
1/2
2
1
1 1
2
p
v
 
 
 

= =
 
 
 
+ +
 
 
 
 
 
 
CASE-I: poor conductor
2
1
1
8

  

 
 
= +
 
 
 
 
Arpan Deyasi
Electromagnetic
Theory

78. 18-12-2021 Arpan Deyasi, EM Theory 78
2
1
1
8
p
v


 

 =
 
 
+
 
 
 
 
2
1
1
1
8
p
v



=
 
 
+
 
 
 
 
2
1 1
1
8
p
v



 
 
= −
 
 
 
 
Maxwell’s equation in conducting medium: Phase velocity
Arpan Deyasi
Electromagnetic
Theory

79. 18-12-2021 Arpan Deyasi, EM Theory 79
Maxwell’s equation in conducting medium: Phase velocity
CASE-II: good conductor
2

 =
2
p
v


 =
2
p
v


=
Arpan Deyasi
Electromagnetic
Theory

80. 18-12-2021 Arpan Deyasi, EM Theory 80
Problem 9
Calculate phase velocity of EM wave travelling at Cu having σ = 5.8⨯10-7 mho/m,
εr = 4, μr = 1.
Soln
2
p
v


=
0
2
p
r
v

  
= 415.22 /sec
p
v m
=
Arpan Deyasi
Electromagnetic
Theory

81. 18-12-2021 Arpan Deyasi, EM Theory 81
Maxwell’s equation in conducting medium: Wavelength
2


=
CASE-I: poor conductor
2
1
1
8

  

 
 
= +
 
 
 
 
2
2
1
1
8



 

 =
 
 
+
 
 
 
 
2
2 1
1
8
 


 
 
 
= −
 
 
 
 
Arpan Deyasi
Electromagnetic
Theory

82. 18-12-2021 Arpan Deyasi, EM Theory 82
Maxwell’s equation in conducting medium: Wavelength
CASE-II: good conductor
2

 =
2
2



 =
2 2


=
Arpan Deyasi
Electromagnetic
Theory

83. 18-12-2021 Arpan Deyasi, EM Theory 83
Electromagnetic wave in a conducting medium shows that there is an exponential
damping or attenuation of the amplitude with distance.
The quantity δ=1/α measures the depth at which e.m wave entering in a
conductor, is damped to 1/e times of its initial amplitude at the surface
This distance is called skin depth.
Maxwell’s equation in conducting medium: Skin depth
Arpan Deyasi
Electromagnetic
Theory

84. 18-12-2021 Arpan Deyasi, EM Theory 84
Maxwell’s equation in conducting medium: Skin depth
Relation between skin depth and attenuation constant
Let the wave attenuation be represented as
( )
0 exp
E E z

= −
At z = α
( )
0 exp
E E 
= −
( )
0
0 exp
E
E
e

= −
1


=
Arpan Deyasi
Electromagnetic
Theory

85. 18-12-2021 Arpan Deyasi, EM Theory 85
Problem 10
For sea-water, σ=5 mho/m, εr = 80. What is the distance an EM wave can be
transmitted at 25 KHz & 25 MHz when the range corresponds to 90% of
maximum?
Soln
( )
exp 0.1
z

− =
2.3
z

=
Arpan Deyasi
Electromagnetic
Theory

86. 18-12-2021 Arpan Deyasi, EM Theory 86
1/2
2
1 1
1 1
2 2

  

 
 
 
= + −
 
 
 
 
 
 
25 , 0.702
At f KHz 
= =
25 , 21.96
At f MHz 
= =
0.702, 3.27
For z m
 = =
21.96, 0.104
For z m
 = =
Arpan Deyasi
Electromagnetic
Theory

87. 18-12-2021 Arpan Deyasi, EM Theory 87
Maxwell’s equation in conducting medium: Skin depth
CASE-I: poor conductor
1
2

 

=
1 2 

  
= =
Arpan Deyasi
Electromagnetic
Theory

88. 18-12-2021 Arpan Deyasi, EM Theory 88
Maxwell’s equation in conducting medium: Skin depth
2

 =
CASE-II: good conductor
1 2

 
= =
Arpan Deyasi
Electromagnetic
Theory

89. 18-12-2021 Arpan Deyasi, EM Theory 89
Problem 11
Calculate the skin depth for radio waves of 3 m wavelength in copper.
Given: 7 7
0
6 10 s/m; 4 10 H/m
−
 =   = 
Soln
Cu is good conductor
2
 =

7 7
2
4 10 6 10
−
 =
   
Arpan Deyasi
Electromagnetic
Theory

90. 18-12-2021 Arpan Deyasi, EM Theory 90
8
3
24 2 3 10
 =
  
6
6.499 10−
 = 
6.499 m
 = 
Arpan Deyasi
Electromagnetic
Theory

91. 18-12-2021 Arpan Deyasi, EM Theory 91
Maxwell’s equation in conducting medium: Skin effect
The phenomenon of high frequency fields, and hence, currents are confined
within a small region of conducting medium inside the surface is known as
skin effect.
This effect becomes more pronounced as frequency increases, and has the result
that the resistance of wire increases with frequency, i.e., effective cross-section of
wire decreases.
For high frequency application, therefore, it is recommended to use a wire
comprised of many fine strands, rather than a single large diameter wire.
Arpan Deyasi
Electromagnetic
Theory