5 slides

Dr. Rakhesh Singh Kshetrimayum
5. Plane Electromagnetic Waves
Dr. Rakhesh Singh Kshetrimayum
3/19/20141 Electromagnetic FieldTheory by R. S. Kshetrimayum

5.1 Introduction
Electromagnetic
Waves
Plane waves Poynting
vector
Plane waves in
various media
3/19/2014Electromagnetic FieldTheory by R. S. Kshetrimayum2
Polarization Lossless
medium
Good
conductor
Fig. 5.1 PlaneWaves
Lossy
conducting
medium
Good
dielectric

5.2 Plane waves
5.2.1What are plane waves?
What are waves?
Waves are a means for transferring energy or information
from one place to another
What are EM waves?What are EM waves?
Electromagnetic waves as the name suggests, are a means for
transferring electromagnetic energy
Why it is named as plane waves?
Mathematically assumes the following form
3/19/20143 Electromagnetic FieldTheory by R. S. Kshetrimayum
( ) ( )
0,
j k r t
F r t F e
ω• −
=
r rr rr

5.2 Plane waves
where is the wave vector and it points in the direction of
wave propagation,
is the general position vector,
ω is the angular frequency, and
is a constant vector
k
r
r
r
F
r
is a constant vector
denotes either an electric or magnetic field ( F is a
notation for field not for the force)
For example, in electromagnetic waves, is either vector
electric ( ) or magnetic field ( )
0F
0F
r
0F
r
0E
r
0H
r

5.2 Plane waves
In rectangular or Cartesian coordinate system
x y zk k x k y k z= + +
r ) ) )
r xx yy zz= + +
r ) ) )
( ) ( ) ( )
22 22 2
k k k k k k ω µε⇒ = • = + + =
r r
Note that the constant phase surface for such waves
( ) ( ) ( )
22 22 2
x y zk k k k k k ω µε⇒ = • = + + =
( ) ( ) tanx y z x y zk r k x k y k z xx yy zz k x k y k z con t• = + + • + + = + + =
r r ) ) ) )) )

5.2 Plane waves
defines a plane surface and hence the name plane waves
Since the field strength is uniform everywhere it is also
known as uniform plane waves
A plane wave is a constant-frequency wave whose
wavefronts (surfaces of constant phase) are infinite parallel
wavefronts (surfaces of constant phase) are infinite parallel
planes
of constant amplitude normal to the phase velocity vector
For plane waves from the Maxwell’s equations,
the following relations could be derived (see Example 4.3)
; ; 0; 0k E H k H E k E k Hωµ ωε× = × = − • = • =
r r r rr r r r r r

5.2 Plane waves
Properties of a uniform plane wave:
Electric and magnetic field are perpendicular to each other
No electric or magnetic field in the direction of propagation
(Transverse electromagnetic wave:TEM wave)
The value of the magnetic field is equal to the magnitude of
The value of the magnetic field is equal to the magnitude of
the electric field divided by η0 (~377 Ohm) at every instant
(magnetic field amplitude is much smaller than the electric field
amplitude)

5.2 Plane waves
The direction of propagation is in the same direction as
Poynting vector
The instantaneous value of the Poynting vector is given by
E2/η0, or H2η0
The average value of the Poynting vector is given by E2/2η0,
The average value of the Poynting vector is given by E2/2η0,
or H2η0/2
The stored electric energy is equal to the stored magnetic
energy at any instant

5.2 Plane waves
5.2.2Wave polarization
Polarization of plane wave refers to the orientation of electric
field vector,
which may be in fixed direction or
may change with time
may change with time
Polarization is the curve traced out by the tip of the arrow
representing the instantaneous electric field
The electric field must be observed along the direction of
propagation

5.2 Plane waves
Types of
polarization
Linear polarized Circularly Elliptically
Linear polarized
(LP)
Circularly
polarized (CP)
Elliptically
polarized (EP)
RHCPLHCP RHEP LHEP

5.2 Plane waves
If the vector that describes the electric field at a point in
space
varies as function of time and
is always directed along a line
which is normal to the direction of propagation
which is normal to the direction of propagation
the field is said to be linearly polarized
If the figure that electric field trace is a circle (or ellipse),
then, the field is said to be circularly (or elliptically) polarized

5.2 Plane waves
Besides, the figure that electric field traces is circle and
anticlockwise (or clockwise) direction,
then, electric field is also said to be right-hand (or left-hand)
circularly polarized wave (RHCP/LHCP)
Besides, the figure that electric field traces is ellipse and
Besides, the figure that electric field traces is ellipse and
anticlockwise (or clockwise) direction,
then, electric field is also said to be right-hand (or left-hand)
elliptically polarized (RHEP/LHEP)

5.2 Plane waves
Let us consider the superposition of
a x- linearly polarized wave with complex amplitude Ex and
a y- linearly polarized wave with complex amplitude Ey,
both travelling in the positive z-direction
Note that E and E may be varying with time for general
Note that Ex and Ey may be varying with time for general
case
so we may choose it for a particular instant of time
Note that since the electric field is varying with both space
and time

5.2 Plane waves
Easier to analyze at a particular instant of time first
And add the time dependence later
The total electric field can be written as
( ) ( ) ( ) zjj
y
j
x
zj
yx eyeExeEeyExEzE yx βφφβ −−
+=+= ˆˆˆˆ 00
r
Note Ex and Ey may be complex numbers and
Ex0 and Ey0 are the amplitudes of Ex and Ey
( ) ( ) ( )yxyx eyeExeEeyExEzE 00

5.2 Plane waves
and are the phases of Ex and Ey
Putting in the time dependence and taking the real part, we
have,
A number of possibilities arises:
( ) ( ) ( )yztExztEtzE yyxx
ˆcosˆcos, 00 φβωφβω +−++−=
r
xφ yφ
A number of possibilities arises:
Linearly polarized (LP) wave:
If both Ex and Ey are real (say Ex = Eox and Ey = Eoy), then,
( ) ( ) ( ) zj
yx
zj
yxLP eyExEeyExEzE ββ −−
+=+= ˆˆˆˆ 00
r

5.2 Plane waves
have,
The amplitude of the electric field vector is given by
( ) ( ) ( )yztExztEtzE yxLP
ˆcosˆcos, 00 βωβω −+−=
r
which is a straight line directed at all times along a line
that makes an angle θ with the x-axis given by the following
relation
01 1
0
tan tan
y y
LP
x x
E E
E E
θ − −
   
= =   
   
( ) ( ) ( ) ( )ztEEtzE yxLP βω −+= cos,
2
0
2
0
r

5.2 Plane waves
If Ex ≠ 0 and Ey = 0,
we have a linearly polarized plane wave in x- direction
( ) ( )xztEtzE oxLP
ˆcos, βω −=
r

5.2 Plane waves
Easier to fix space to see the polarization
For a fixed point in space (say z=0),
For all times, electric field will be directed along x-axis
( ) ( )xtEtzE ox
z
LP
ˆcos,
0
ω=
=
r
For all times, electric field will be directed along x-axis
hence, the field is said to be linearly polarized along the x-
direction

5.2 Plane waves
Fig. 5.2 (a) LP wave

5.2 Plane waves
Circularly polarized (CP) wave:
Now consider the case Ex = j Ey = Eo, where Eo is real so
that
0 2
0 0; ;
j
j
x yE E e E E e
π
−
= =
The time domain form of this field is (putting in the time
dependence and taking the real part)
0 0x y
ˆ ˆ( ) j z
RHCP oE E x jy e β−
= −
r
ˆ ˆ( , ) [ cos( ) cos( )]
2
RHCP oE z t E x t z y t z
π
ω β ω β= − + − −
r

5.2 Plane waves
Note that x- and y-components of the electric field have the
same amplitude
but are 900 out of phase
Let us choose a fixed position (say z=0), then,
which shows that the polarization rotates with
uniform angular velocity ω in anticlockwise direction
for propagation along positive z-axis
( )1 1sin
tantan tan
cos
RHCP
t
t t
t
ω
θ ω ω
ω
− − 
 = = =   
 

5.2 Plane waves
An observer sitting at z=0 will see
the electric field rotating in a circle and
the field never goes to zero
Since the fingers of right hand point in the direction of
rotation of the tip of the electric field vector
rotation of the tip of the electric field vector
when the thumb points in the direction of propagation,
this type of wave is referred to as right hand circularly
polarized wave (RHCP wave)

5.2 Plane waves
Fig. 5.2 (b) RHCP wave
x
y

5.2 Plane waves
Elliptically polarized (EP) wave:
Now, consider a more general case of EP wave,
when the amplitude of the electric field in the x- and y-
directions are not equal in
amplitude and
amplitude and
phase
unlike CP wave, so that,
have,
( ) ( ) zjj
EP eyAexzE βφ −
+= ˆˆ
r
( ) ( ) ( )yztAxzttzE EP
ˆcosˆcos, φβωβω +−+−=
r

5.2 Plane waves
If φ is in the upper half of the complex plane
then the wave is LHEP
whereas φ is in the lower half of the complex plane,
then the wave is RHEP
Let us choose a fixed position (say z=0) like in the CP case,
Let us choose a fixed position (say z=0) like in the CP case,
then,
Some particular cases:
( ) ( )0
ˆ ˆcos cosEP z
E t x A t yω ω φ
=
= + +
r

5.2 Plane waves
( )( ) ( )
( )( ) ( )
{ }
0
0
0
0
ˆ ˆ( ) 1, 0; cos
ˆ ˆ( ) 1, ; cos
z
z
a A E E t x y LP
b A E E t x y LP
φ ω
φ π ω
π
=
=
= = = +
= = = −
r
r
r
( ) ( ){ } ( )
( ) ( ){ } ( )
0
0
0
0
ˆ ˆ( ) 1, ; cos sin
2
ˆ ˆ( ) 1, ; cos sin
2
z
z
c A E E t x y t LHCP
d A E E t x y t RHCP
π
φ ω ω
π
φ ω ω
=
=
= = = −
= = − = +
r
r

5.2 Plane waves
( ) ( ){ } ( )
( ) ( ){ } ( )
( ) ( )
0
0
0
0
0
ˆ ˆ( ) 3, ; cos 3sin
2
ˆ ˆ( ) 0.5, ; cos 0.5sin
2
ˆ ˆ( ) 1, ; cos cos
4 4
z
z
e A E E t x y t LHEP
f A E E t x y t RHEP
g A E E t x y t LHEP
π
φ ω ω
π
φ ω ω
π π
φ ω ω
=
=
= = = −
= = − = +
  
= = = + +  
 
r
r
r
( ) ( )
( ) ( )
0
0
0
0
ˆ ˆ( ) 1, ; cos cos
4 4
ˆ ˆ( ) 1, 3 ; cos cos 3
4 4
z
z
g A E E t x y t LHEP
h A E E t x y t RHEP
φ ω ω
π π
φ ω ω
=
=
= = = + +  
  
  
= = − = + −  
  
r

5.2 Plane waves
Fig. 5.2 (c) LHEP wave
Direction of propagation
Electric field
Magnetic field at each point is orthogonal to the electric field
x
y

5.3 Poynting vector & power flow in EM fields
The rate of energy flow per unit area in a plane wave is
described by a vector termed as Poynting vector
which is basically curl of electric field intensity vector and
magnetic field intensity vector
The magnitude of Poynting vector is the power flow per unit
area and
it points along the direction of wave propagation vector
*
S E H= ×
r r r

The average power per unit area is often called the intensity
of EM waves and it is given by
Let us try to derive the point form of Poynting theorem from
( )*1
Re
2
avgS E H= ×
r r r
Let us try to derive the point form of Poynting theorem from
two Maxwell’s curl equations
t
H
E
∂
∂
−=×∇
r
r
µ J
t
E
H
r
r
r
+
∂
∂
=×∇ ε

From vector analysis,
We can further simplify
)()()()()( J
t
E
E
t
H
HHEEHHE
r
r
r
r
rrrrrrr
+
∂
∂
•−
∂
∂
−•=×∇•−×∇•=×•∇ εµ
( )AA
tt
A
A
rr
r
r
Q •
∂
∂
=
∂
∂
•
2
1
Basically a point relation
It should be valid at every point in space at every instant of time
( ) ( ) ( )
2 2
E H H H E E E J
t t
µ ε∂ ∂
∴∇ • × = − • − • − •
∂ ∂
r r r r r r r r
( )AA
tt
AQ •
∂
=
∂
•
2

The power is given by the integral of this relation of Poynting
vector over a volume as follows
We can interchange the volume integral and partial
∫∫∫ •−•
∂
∂
−•
∂
∂
−=
VVV
dvJEdvEE
t
dvHH
t
rrrrrr
)(
2
)(
2
εµ
( ) ( ) ∫∫∫ •=•×=×•∇
SSV
sdSsdHEdvHE
rrrrrrr
We can interchange the volume integral and partial
derivative w.r.t. time
∫∫∫∫ −
∂
∂
−
∂
∂
−=•
VVVS
dvEdvE
t
dvH
t
sdS 222
2
1
2
1
σεµ
rr

This is the integral form of Poynting vector and power flow
in EM fields
Poynting theorem states that
the power coming out of the closed volume is equal to
the total decrease in EM energy per unit time i.e. power loss
the total decrease in EM energy per unit time i.e. power loss
from the volume which constitutes of
rate of decrease in magnetic energy stored in the volume
rate of decrease in electric energy stored in the volume
Ohmic power loss (energy converted into heat energy per unit
time) in the volume

Now going back to the last four points of plane waves:
The direction of propagation is in the same direction as of
Poynting vector
The instantaneous value of the Poynting vector is given by
E2/η0, or H2η0
E / 0, or H 0
The average value of the Poynting vector is given by E2/2η0, or
H2η0/2
The stored electric energy is equal to the stored magnetic
energy at any instant

Let us assume a plane wave traveling in the +z direction in
free space, then
The instantaneous value of the Poynting vector:
0 0
0 0
0
;jk zj z j zz E
E E e E e H eβ β
η
−− −×
= = =
r)r r r r
The instantaneous value of the Poynting vector:
( ) ( ) ( )
( ) ( ) ( )
0
2
0
0
00
0
0000
00
00
0
0
ˆˆˆˆ
ˆ
1ˆ
ηηη
ηη
ββ
zEEEzzEEEEz
EzEe
Ez
eEHES zjzj
rrrrrrr
rr
r
rrrr
=
•
=
•−•
=
××=






 ×
×=×= −∗

o Note that the direction of Poynting vector is also in the z-
direction same as that of the wave vector
o The average value of the Poynting vector:
( )
2
0
2
0 ˆˆ
Re
1
Re
1 zEzE
HESavg
rr
rrr
=





=×= ∗
o Stored Electric Energy:
o Stored Magnetic Energy:
( )
00 2
Re
2
Re
2 ηη
HESavg =






=×=
2
0
1
2
ew Eε=
2
2 2 20
0 0 0 02
0 0
1 1 1 1
2 2 2 2
m e
E
w H E E w
ε
µ µ µ ε
η µ
= = = = =

5.4 Plane waves in various media
A media in electromagnetics is characterized by three parameters:
ε, µ and σ
5.4.1 Lossless medium
In a lossless medium,
ε and µ are real, σ=0, so β is real ( )ωεσωµγ jj +=2
Q
ε and µ are real, σ=0, so β is real
Assume the electric field with
only x- component,
no variation along x- and y-axis and
propagation along z-axis, i.e.,
0
E E
x y
∂ ∂
= =
∂ ∂
r r
( )ωεσωµγ jj +=Q
( ) µεωββµεωγ =⇒==
2222
jj

Helmholtz wave equation reduces to
whose solution gives waves in one dimension as follows
02
2
2
=+
∂
∂
xx EE
z
β
where E+ and E- are arbitrary constants
e j z j z
xE E E eβ β+ − − +
= +

Putting in the time dependence and taking real part, we get,
For constant phase,
ωt-βz=constant=b(say)
)cos()cos(),( ztEztEtzEx βωβω ++−= −+
ωt-βz=constant=b(say)
Since phase velocity,
0 0
) 1 1
( )p
r r
dz d t b
v
dt dt
ω ω
β β µε µ µ ε ε
−
= = = = =
β ω µε=Q

For free space,
which is the speed of light in free space
This emergence of speed of light from electromagnetic
smcv p /103
1 8
00
×===
εµ
This emergence of speed of light from electromagnetic
considerations is one of the main contributions from
Maxwell’s theory
The magnetic field can be obtained from the source free
Maxwell’s curl equation

HjE
rr
ωµ−=×∇
( )
ˆ ˆ ˆ
ˆ e j z j z
x y z
E j E j j
H y E E eβ β+ − − +∇× ∇× ∂ ∂ ∂ ∂ 
= − = = = + 
r r
r
( )ˆ e
e 0 0j z j z
H y E E e
j x y z z
E E eβ β
ωµ ωµ ωµ ωµ
+ − − +
= − = = = + 
∂ ∂ ∂ ∂ 
+
( )
{ }( )
{ }
( ) ( )( ) ( )
ˆ ˆ
( ) ( ) 1
ˆ ˆ[ ]
j z j zj z j z
j z j z
j z j z
j E e E ej E e E e j
H j y j y
E e E e
y E e E e y
β ββ β
β β
β β
ββ β
ωµ ωµ
β
ωµ η
+ − − ++ − − +
+ − − +
+ − − +
− −− +
= =
−
= = −
r

η is the wave impedance of the plane wave
For free space,
Hy
Ex
===
ε
µ
β
ωµ
η
5.4.2 Lossy conducting medium
If the medium is conductive with a conductivity σ, then the
Maxwell’s curl equations can be written as
Ω=== 377120π
ε
µ
η
o
o
o

;E j Hωµ∇× = −
r r
( ) ;effH j E E j E j Eωε σ ωε σ ωε∇ × = + = + =
r r r r r
The effect of the conductivity has been absorbed in the
complex frequency dependent effective permittivity
( ) 1eff
j j
j
σ σ σ
ε ω ε ε ε
ω ω ωε
 
= + = − = − 
 

We can define a complex propagation constant
( ) ( )
22 2 2
0effE E E j Eω µε ω γ⇒ ∇ + = ∇ + =
r r r r
( )effj jγ ω µε ω α β= = +
where α is the attenuation constant and β is the phase
constant

What is implication of complex wave vector?
The wave is exponentially decaying (see example 4.4).
The dispersion relation for a conductor (usually non-
magnetic) is
( )ε ω ω
where neff is the complex refractive index
( )
( )
( ) ( )0 0 0 0
0
eff
eff eff effj j j n j n
c
ε ω ω
γ ω µε ω ω µ ε ω µ ε ω ω
ε
= = = =

1-D wave equation for general lossy medium becomes
whose solution is 1-D plane waves as follows
02
2
2
=−
∂
∂
x
x
E
z
E
γ
zjzzjzzz
x eeEeeEeEeEzE βαβαγγ −−−++−−+ +=+=)(

Putting the time dependence and taking real part, we get,
The magnetic field can be found out from Maxwell’s
equations as in the previous section
)cos()cos(),( zteEzteEtzE zz
x βωβω αα ++−= −−+
equations as in the previous section
1
( ) [ ]z z
y
eff
H z E e E eγ γ
η
+ − −
= −

where useful expression for intrinsic impedance is
The electric field and magnetic field are no longer in phase as
( ) ( )
0 0 0
0
eff
effeff
j j
j
ωµ ωµ µ
η
γ ε ωω µ ε ω
= = =
The electric field and magnetic field are no longer in phase as
εeff is complex
Poynting vector or power flow for this wave inside the lossy
conducting medium is

it is decaying in terms of square of an exponential function
5.4.3 Good dielectric/conductor
2*
2* 2
* *
ˆ ˆ ˆ ˆ
eff eff
z j z z j z
z j z z j z z
eff
EE e e e e
S E H E e e x y E e e z e z
α β α β
α β α β α
η η η
++ − − − +
+ − − + − − −
 
= × = × = × =  
 
r r r
5.4.3 Good dielectric/conductor
Note that σ/ωε is defined as loss tangent of a medium
A medium with σ/ωε <0.01 is said to be a good insulator
whereas a medium with σ/ωε >100 is said to be a good
conductor

For good dielectric,
can be approximated usingTaylor’s series expansion obtain α
and β as follows:
( 1 )
j
w j
σ
σ ε γ ω µε
ωε
<< ∴ = −Q
ε
µσ
α
2
= µεωβ =
For a good conductor,
Therefore,
ε
α
2
= µεωβ =
ωεσ >>
22
)1(
µσ
βα
ωµσ
γ
w
j ==⇒+≅

Skin effect
The fields do attenuate as they travel in a good dielectric
medium
α in a good dielectric is very small in comparison to that of a
good conductor
good conductor
As the amplitude of the wave varies with e-αz,
the wave amplitude reduces its value by 1/e or 37% times
over a distance of
1
δ
α
=
1 2 2 1
2 f fβ ωµσ π µσ π µσ
= = = =

which is also known as skin depth
This means that in a good conductor
(a) higher the frequency, lower is the skin depth
(b) higher is the conductivity, lower is the skin depth and
(c) higher is the permeability, lower is the skin depth
(c) higher is the permeability, lower is the skin depth
Let us assume an EM wave which has x-component and
traveling along the z-axis
Then, it can be expressed as
( ) ( )tzjz
x eeEtzE ωβα −−−
= 0,

Taking the real part, we have,
Substituting the values of α and β for good conductors, we
have,
( ) ( )zteEtzE z
x βωα
−= −
cos, 0
have,
Now using the point form of Ohm’s law for conductors, we
can write
( ) ( )zfteEtzE zf
x µσπωµσπ
−= −
cos, 0
( ) ( )zfteEtzEJ zf
xx µσπωσσ µσπ
−== −
cos, 0

What is the phase velocity and wavelength inside a good
conductor?
2
; 2pv
ω π
ωδ λ πδ
β β
= = = =

5.5 Summary
Electromagnetic
Waves
Plane waves
Polarization
Lossless
Good
conductor
Plane waves in
various media
Lossy
conducting
Good
dielectric
( ) ( ) tanx y z x y zk r k x k y k z xx yy zz k x k y k z con t• = + + • + + = + + =
r r ) ) ) )) )
Poynting vector
Lossless
medium
conductor
Fig. 5.3 Plane waves in a nutshell
conducting
medium
dielectric
ˆ ˆ( ) j z
RHCP oE E x jy e β−
= −
r
∫∫∫∫ −
∂
∂
−
∂
∂
−=•
VVVS
dvEdvE
t
dvH
t
sdS 222
2
1
2
1
σεµ
rr
( ) ( ) ( )
2 2
E H H H E E E J
t t
µ ε∂ ∂
∴∇ • × = − • − • − •
∂ ∂
r r r r r r r r
µεωβ =
µεβ
ω 1
==pv
ε
µ
β
ωµ
η ==
( ) 





−=
ωε
σ
εωε
j
eff 1
( )ωε
µ
γ
ωµ
η
eff
eff
j 00
==
ε
µσ
α
2
=
µεωβ =
µσπβα
δ
f
111
===
2
ωµσ
βα ==
2
; 2pv
ω π
ωδ λ πδ
β β
= = = =
( ) ( )zfteEtzEJ zf
xx µσπωσσ µσπ
−== −
cos, 0
( ) ( ) zj
yxLP eyExEzE β−
+= ˆˆ 00
r
( ) ( ) zjj
EP eyAexzE βφ −
+= ˆˆ
r

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