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Plane electromagnetic waves
Plane electromagnetic Waves
 We will assume that the vectors for
the electric and magnetic fields in an
electromagnetic wave have a specific
space-time behavior that is consistent
with Maxwell’s equations.
 Assume an em wave that travels in
the x direction with E and B as
shown
Poynting Vector
 Electromagnetic waves carry energy
 As they propagate through space, they can transfer that energy to
objects in their path
 The rate of flow of energy in an electromagnetic wave is described by
a vector, S, called the Poynting vector
 The Poynting vector is defined as
1
S  E B  E  H
μo
 Its direction is the direction of propagation
EB
Poynting Vector
 For a plane-polarized em wave, the direction of Poynting vector is
ˆ
S  E  H  ˆjE y  kH
z
 Ey Hz ( ˆj  kˆ )
 Ey Hzˆi
 This is time dependent
 Its magnitude varies in time
 The magnitude of S represents the rate at which energy flows
through a unit surface area perpendicular to the direction of the
wave propagation
 This is the power per unit area
 The SI units of the Poynting vector are J/(s.m2) = W/m2
Intensity
 The wave intensity, I, is the time average of S (the Poynting
vector) over one or more cycles
 When the average is taken, the time average of cos2(kx - ωt)
= ½ is involved
I  Savg
2
max
2
max
Emax Bmax E
cB



2 μo
2 μo c
2 μo
Energy Density associated with em waves
 The energy density, u, is the energy per unit volume.
 For the electric field, uE= ½ εoE2
 For the magnetic field, uB = ½ μoB2
 Since B = E/c and c  1
μoεo
2
1
B
uB  uE  εoE 2 
2
2μo
Energy Density
 The instantaneous energy density associated with the magnetic field
of an em wave equals the instantaneous energy density associated
with the electric field
 In a given volume, the energy is shared equally by the two fields
 The total instantaneous energy density is the sum of the
energy densities associated with each field
 u =uE + uB = εoE2 = B2 / μo
 When this is averaged over one or more cycles, the total average
becomes
 uavg = εo(E2)avg = ½ εoE2max = B2max / 2μo
 In terms of I, I = Savg = cuavg
 The intensity of an em wave equals the average energy density
multiplied by the speed of light
Poynting theorem
 Consider an elementary area da in the space in which em waves are
propagating.
 If S is the Poynting vector, then it gives the time rate of flow of energy
per unit area and is called POWER FLUX through unit area.
 The power flux through the area dA will be
dP  S .dA
 The total power through the closed surface is given by
P 
 S .dA
using Gauss's divergence theorem
P   .SdV
...(1)
where .S gives the amount of energy diverging per second out of
enclosed volume of the space enclosed by the closed surface.
Cont.
 From Maxwell equations
B
H
 E  
 
...(2)
t
t
D
E
 H  J 
 J 
...(3)
t
t
 The Poynting vector is given as
S  EH
Taking divergence
.S  .( E  H )
(Using idenetity .( A  B)  B.(  A)  A.(  B)
.S  H.(   E)  E.(   H) ...(4)
Cont.
 Taking dot product of equation (2) with H
H
 ( H .H )
=-
t
t
1 H 2
 H .(  E )  - 
...(5)
2
t
H .(  E )    H .
 Taking dot product of equation (3) with E
D
E
E.(  H )  E.J  E.
 J .E   E.
t
t
1 E 2
 E.(  H )=J .E  
...(6)
2 t
 Using equation (5) and (6) in (4)
.S  H.(   E)  E.(   H)
1 H 2
1 E 2
.S  - 
 J .E  
2
t
2 t
1 H 2 1 E 2
 .S  
 
  J .E
2
t
2 t
 1
1

 .S    H 2   E 2    J .E
t  2
2

....(7)
Cont.
 Integrating equation (7) over the volume V bounded by the surface
 .SdV  
V
V
 1
1 2
2

H

 E  dV    J .EdV

t  2
2

V
using Gauss's divergence theorem for first term
 S .dA  
A
V
 1
1 2
2

H

 E  dV    J .EdV

t  2
2

V
....(8)
 The first term represents flow of energy per unit time i.e. power
flux
 Second term corresponds to rate of change of total energy
 The term on R.H.S. represents the work done by the filed on the
source in setting up a current
Cont.
 Energy associated with the electric field
 E2
UE 
2
 and that of the magnetic field
 H 2 B2
Um 

2
2
 The total energy stored will be
 E2 H 2
U  Ue  Um 

2
2
Cont.
 Therefore equation (8) can be written as
u
dV   J .EdV
A S .dA  
t
V
V
....(9)
 The equation (9) represents Poynting theorem.
 It states that the sum of flow of energy per unit volume
across the boundary of the surface and time rate of
change of electromagnetic energy is equal to the work
done by the electromagnetic field on the source.
Cont.
 In free space, J=0. Therefore
 S.dA  
A
V
u
dV  0 ....(10)
t
 From equation (10), we can say that the power flux
through a closed area is equal to the rate of outflow of
energy from the volume enclosed by that area.
 Equation of continuity:
using
 S .dA   .SdV
A
V
u
.SdV   dV  0

t
V
V
 .S 
u
0
t
Electromagnetic wave in a medium with finite
permeability, permittivity and conductivity (Conducting
medium)
 For plane-polarized em waves
 2 Ey
 
E y
 
 2 Ey
0
...(1)
x
t
t
2 H z
H z
2 H z
and
 
 
 0 ...(2)
2
2
x
t
t
1
The product   2 (wave velocity)
v
and  is called Magnetic diffusivity
2
2
 The second term is called diffusion term and is due to finite
conductivity (or conduction currents) of the medium.
 For perfect insulators, this term is absent from em wave equation.
Solution of em wave equation for a conducting
medium
 Let the solution of wave equation is given by
E y  Eo e(iwt  x )
...(3)
and H y  H o e(iwt  x )
...(4)
 Substituting the eq. (3) in eq. (1)
 2 Eo e(iwt  x )   i Eo e(iwt  x )   i 2 2 Eo e(iwt  x )  0
  2   i   2  0
  2   i   2
  2   ( i   )
For good conductors,   
  2   i
...(5)
Cont.
 For good conductors
  2   i
2i 

2 

 (1  1  2i)
 (1  i)
2
2
2
  
1/2
   (1  i ) 

2


   (1  i )k
...(6)
  
where k  

2


1/2
Cont.
 Therefore from equation (3)
E y  Eo e
( iwt  x )
E y  Eo e
( iwt  (1 i ) kx )
 Taking the negative sign, which gives the wave propagating in the positive
x-direction
E y  Eo e(iwt (1i ) kx )
 E y  Eo e kx e(iwt ikx )
 kx i ( wt  kx )
 E y  Eo e e
...(7)
 kx
 This equation represent a progressive wave having amplitude equal to Eo e
 The amplitude of the wave goes on decreasing as the wave propagates
deeper into the medium.
Cont.
 Therefore from equation (3)
E y  Eo e(iwt  x )
E y  Eo e(iwt (1i ) kx )
 Taking the negative sign, which gives the wave propagating in the
positive x-direction
E y  Eo e(iwt (1i ) kx )
 E y  Eo e kx e(iwt ikx )
 kx i ( wt  kx )
 E y  Eo e e
...(7)
Skin depth (d)
 The amplitude of the em wave decreases exponentially with
distance of penetration of wave. The decrease in the amplitude or
the attenuation of the field vector is expressed in terms of skin
depth.
 It is defined as the distance beyond the surface of the conductor
inside it at which the amplitude of the field vector is reduced to
1/e of its value at the surface.
 Let amplitude at a depth x is Ex.
 Then
Ex  Eo e  kx
 If skin depth is d then
1
Ed  Eo
where Eo is the amplitude at thee surface and Ed is the amplitude at
depth d
Skin depth (d)
1
Ed  Eo
e
Ed 1

  e 1
Eo e
Eo e  kd

 e 1
Eo
 e  kd  e 1
 kd  1
1/2
1  2 
d  

k   
 It shows that the spin depth varies inversely as the square root of
the conductivity of the medium and the frequency of em waves.
Dispersion of em waves in a conductor
 The dispersion of waves occur when their group and phase velocity are
not equal.
 The equation for the electric waves through a conducting medium is
given by
Ey  Eoekx ei ( wt kx )
 The phase velocity of the wave will be
v

k
where   2 , k  2 
Also k 
1
d
 v  2d
If c is wavelength in the conductor then
v  c
c
 c  2d  c  2d  d 
2
The skin depth depends
on wavelength.
Dispersion of em waves in a conductor
 The phase velocity of the em wave is given as
1
v
and for free space, c 

Also  r =
1
 0 0





 o  o r o o
c2
 r =
r v 2
but for conductors r  1
c2
 r = 2
v
  r v 2 =c 2
Differentiating w.r.t. 
d
dv
2 r v
 v2 r  0
d
d
dv
v d r


d
2 r d 
Dispersion of em waves in a conductor
The group velocity and phase velocity are related as
dv
vg  v  
d
v d r
 vg  v 
2 r d 

 d r 
 vg  v  1 

2

d

r


 vg  v
 So the em waves suffer anomalous dispersion in the
conductors.
Characteristics impedance of a medium to
the EM waves
 Characteristic impedance,
Z=(Instantaneous electric vector)/(Instantaneous magnetic
vector)
=Ey/Hz
Characteristics impedance of a dielectric
medium to the EM waves
 In the dielectric medium,  =0
 The wave equation for the electric and magnetic vectors are:
E y  E0 e i ( wt  kx )
H z  H 0 e i ( wt  kx )
 Characteristic impedance,
Z=(Instantaneous electric vector)/(Instantaneous magnetic
vector)
=Ey/Hz