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5. Plane Electromagnetic Waves
Dr. Rakhesh Singh Kshetrimayum
1
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.1 Introduction
Electromagnetic
Waves
Poynting
vector
Plane waves
Polarization
Lossless
medium
Lossy
conducting
medium
Plane waves in
various media
Good
conductor
Good
dielectric
Fig. 5.1 Plane Waves
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
5.2.1 What are plane waves?
What are waves?
Waves are a means for transferring energy or information
from one place to another
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
r r
r j ( kr •rr −ωt )
F ( r , t ) = F0 e
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
r
where k is the wave vector and it points in the direction of
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wave propagation,
r
r is the general position vector,
ω is the angular frequency, and
r
F 0 is a constant vector
r
F 0 denotes either an electric or magnetic field ( F is a
notation for field not for the force)
r
For example, in electromagnetic waves, F 0 is either vector
r
r
electric ( E 0 ) or magnetic field ( H 0 )
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
In rectangular or Cartesian coordinate system
r
)
)
)
k = kxx + ky y + kzz
r
)
) )
r = xx + yy + zz
r r
2
2
2
⇒ k = k • k = ( k x ) + ( k y ) + ( k z ) = ω 2 µε
2
Note that the constant phase surface for such waves
r r
)
)
)
) )
)
k • r = ( k x x + k y y + k z z ) • ( xx + yy + zz ) = k x x + k y y + k z z = con tan t
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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
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)
r r
r r
r r r
r r r
k × E = ωµ H ; k × H = −ωε E ; k • E = 0; k • H = 0
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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 electric field divided by η0 (~377 Ohm) at every instant
(magnetic field amplitude is much smaller than the electric field
amplitude)
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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,
or H2η0/2
The stored electric energy is equal to the stored magnetic
energy at any instant
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
5.2.2 Wave polarization
Polarization of plane wave refers to the orientation of electric
field vector,
which may be in fixed direction or
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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Types of
polarization
Linear polarized
(LP)
LHCP
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Circularly
polarized (CP)
RHCP
Electromagnetic Field Theory by R. S. Kshetrimayum
Elliptically
polarized (EP)
RHEP
LHEP
3/19/2014
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
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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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
anticlockwise (or clockwise) direction,
then, electric field is also said to be right-hand (or left-hand)
elliptically polarized (RHEP/LHEP)
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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 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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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
r
jφ
E (z ) = (E x xˆ + E y yˆ )e − jβz = E x 0 e jφ x xˆ + E y 0 e y yˆ e − jβz
(
)
Note Ex and Ey may be complex numbers and
Ex0 and Ey0 are the amplitudes of Ex and Ey
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
φx and φy are the phases of Ex and Ey
Putting in the time dependence and taking the real part, we
have,
r
E (z , t ) = E x 0 cos(ωt − βz + φ x )xˆ + E y 0 cos(ωt − β z + φ y )yˆ
A number of possibilities arises:
Linearly polarized (LP) wave:
If both Ex and Ey are real (say Ex = Eox and Ey = Eoy), then,
r
ELP ( z ) = (E x xˆ + E y yˆ )e − jβz = (E0 x xˆ + E0 y yˆ )e − jβz
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Putting in the time dependence and taking the real part, we
r
have,
ELP ( z , t ) = E0 x cos(ωt − β z )xˆ + E0 y cos(ωt − βz ) yˆ
The amplitude of the electric field vector is given by
r
ELP (z , t ) =
(E0 x )2 + (E0 y )2 cos(ωt − βz )
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
 Ey
 Ex
θ LP = tan −1 
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Electromagnetic Field Theory by R. S. Kshetrimayum

 E0 y 
−1
=
 tan 

E

 0x 
3/19/2014
5.2 Plane waves
If Ex ≠ 0 and Ey = 0,
we have a linearly polarized plane wave in x- direction
r
ELP ( z , t ) = Eox cos(ωt − βz )xˆ
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Easier to fix space to see the polarization
For a fixed point in space (say z=0),
r
ELP ( z , t )
z =0
= Eox cos(ωt )xˆ
For all times, electric field will be directed along x-axis
hence, the field is said to be linearly polarized along the x-
direction
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Fig. 5.2 (a) LP wave
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Circularly polarized (CP) wave:
Now consider the case Ex = j Ey = Eo, where Eo is real so
that
Ex = E0 e j 0 ; E y = E0 e
−j
π
2
;
r
ERHCP = Eo ( xˆ − jyˆ )e− jβ z
The time domain form of this field is (putting in the time
dependence and taking the real part)
r
π
ERHCP ( z , t ) = Eo [ xˆ cos(ωt − β z ) + yˆ cos(ωt − β z − )]
2
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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,
 sin ωt 
−1
 = tan  tan (ωt )  = ωt
 cos ωt 
θ RHCP = tan −1 
which shows that the polarization rotates with
uniform angular velocity ω in anticlockwise direction
for propagation along positive z-axis
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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
when the thumb points in the direction of propagation,
this type of wave is referred to as right hand circularly
polarized wave (RHCP wave)
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Fig. 5.2 (b) RHCP wave
x
y
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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 ydirections are not equal in
amplitude and
phase
unlike CP wave, so that,
r
EEP (z ) = xˆ + Ae jφ yˆ e − jβz
(
)
Putting in the time dependence and taking the real part, we
have,
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r
E (z , t )EP = cos(ωt − β z )xˆ + A cos(ωt − β z + φ ) yˆ
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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,
then,
r
EEP
z =0
= cos (ωt ) xˆ + A cos (ωt + φ ) yˆ
Some particular cases:
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
(a)
(b)
(c )
(d )
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r
A = 1, φ = 0; E
= E0 cos (ωt )( xˆ + yˆ )
z =0
r
A = 1, φ = π ; E
= E0 cos (ωt )( xˆ − yˆ )
z =0
A = 1, φ =
π
2
A = 1, φ = −
r
; E
π
2
z =0
r
; E
( LP )
( LP )
= E0 {cos (ωt ) xˆ − yˆ sin (ωt )}
z =0
= E0 {cos (ωt ) xˆ + yˆ sin (ωt )}
Electromagnetic Field Theory by R. S. Kshetrimayum
( LHCP )
( RHCP )
3/19/2014
5.2 Plane waves
(e )
(f)
(g)
( h)
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A = 3, φ =
π
2
r
; E
A = 0.5, φ = −
π
z =0
= E0 {cos (ωt ) xˆ − yˆ 3sin (ωt )}
r
; E
( LHEP )
= E0 {cos (ωt ) xˆ + yˆ 0.5sin (ωt )}
( RHEP )
2
r

π
π 

A = 1, φ = ; E
= E0 cos (ωt ) xˆ + yˆ cos  ωt +   ( LHEP )
z =0
4
4 


r

π
π 

A = 1, φ = −3 ; E
= E0 cos (ωt ) xˆ + yˆ cos  ωt − 3   ( RHEP )
z =0
4
4 


z =0
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.2 Plane waves
Fig. 5.2 (c) LHEP wave
Direction of propagation
Electric field
x
Magnetic field at each point is orthogonal to the electric field
y
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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
r r r*
S = E×H
The magnitude of Poynting vector is the power flow per unit
area and
it points along the direction of wave propagation vector
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.3 Poynting vector & power flow in EM fields
The average power per unit area is often called the intensity
of EM waves and it is given by
r
r r*
1
S avg = Re E × H
2
(
)
Let us try to derive the point form of Poynting theorem from
two Maxwell’s curl equations
r
r
∂H
∇ × E = −µ
∂t
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Electromagnetic Field Theory by R. S. Kshetrimayum
r
r
∂E r
∇× H =ε
+J
∂t
3/19/2014
5.3 Poynting vector & power flow in EM fields
From vector analysis,
r
r
r r
r
r
r
r
r
r
∂H
∂E r
∇ • ( E × H ) = H • (∇ × E ) − E • (∇ × H ) = H • (− µ
) − E • (ε
+ J)
∂t
∂t
We can further simplify
r
r ∂A 1 ∂ r r
Q A•
=
A• A
∂t 2 ∂t
(
)
r r
µ ∂ r r ε ∂ r r r r
∴∇ • ( E × H ) = −
(H • H ) −
(E • E) − E • J
2 ∂t
2 ∂t
Basically a point relation
It should be valid at every point in space at every instant of time
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.3 Poynting vector & power flow in EM fields
The power is given by the integral of this relation of Poynting
vector over a volume as follows
r r
r r
r r
r
µ ∂ r r
ε
=
−
(
H
•
H
)
dv
−
∇
•
E
×
H
dv
=
E
×
H
•
d
s
=
S
•
d
s
∫
∫
∫
∫
(
V
)
(
S
)
S
2 ∂t
V
r r
∂ r r
( E • E )dv − ∫ E • Jdv
2 ∫ ∂t
V
V
We can interchange the volume integral and partial
derivative w.r.t. time
r r
∂ 1
∂ 1 2
2
2
S
•
d
s
=
−
µ
H
dv
−
ε
E
dv
−
σ
E
dv
∫S
∫
∫
∫
∂t V 2
∂t V 2
V
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.3 Poynting vector & power flow in EM fields
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
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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.3 Poynting vector & power flow in EM fields
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
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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.3 Poynting vector & power flow in EM fields
Let us assume a plane wave traveling in the +z direction in
free space, then
r r − jβ z r − jk z r
E = E0 e
= E0e 0 ; H =
) r
z × E0
η0
e− j β z
The instantaneous value of the Poynting vector:
r
r r r∗
r
r − jβz  zˆ × E0 jβz  1 r
S = E × H = E0 e
× 
e  =
E0 × zˆ × E0
 η0
 η0
r 2
r r
r r
r r
E0 zˆ
zˆ E0 • E0 − E0 E0 • zˆ zˆ E0 • E0
=
=
=
(
(
)
)
(
( )(
) (
η0
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Electromagnetic Field Theory by R. S. Kshetrimayum
)
)
η0
η0
3/19/2014
5.3 Poynting vector & power flow in EM fields
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:
r 2  r 2

r
r r ∗ 1  E0 zˆ  E0 zˆ
1
S avg = Re E × H = Re
=

2
2  η0 
2η 0


1
2
w
=
ε
E
o Stored Electric Energy:
e
0
2
o Stored Magnetic Energy:
(
)
1
1 E2 1 ε0 2 1
2
wm = µ0 H = µ0 2 = µ0
E = ε 0 E 2 = we
2
2 η0 2 µ0
2
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
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,
2
ε and µ are real, σ=0, so β is real Q γ = jωµ (σ + jωε )
γ 2 = j 2ω 2 µε = ( jβ )2 ⇒ β = ω µε
Assume the electric field with
only x- component,
no variation along x- and y-axis and
propagation along z-axis, i.e.,
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Electromagnetic Field Theory by R. S. Kshetrimayum
r
r
∂E ∂E
=
=0
∂x ∂y
3/19/2014
5.4 Plane waves in various media
Helmholtz wave equation reduces to
∂2
∂z 2
Ex + β 2 Ex = 0
whose solution gives waves in one dimension as follows
Ex = E + e − j β z + E − e + j β z
where E+ and E- are arbitrary constants
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
Putting in the time dependence and taking real part, we get,
E x ( z , t ) = E + cos(ωt − βz ) + E − cos(ωt + βz )
For constant phase,
ωt-βz=constant=b(say)
Since phase velocity,
dz d ωt − b) ω
vp =
= (
)= =
dt dt
β
β
1
µε
=
1
µ r µ 0ε r ε 0
Q β = ω µε
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
For free space,
vp =
1
µ 0ε 0
= c = 3 × 10 8 m / s
which is the speed of light in free space
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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
r
r
∇ × E = − jωµH
r
r
r
∇ × E j∇ × E
j
H =−
=
=
jωµ
ωµ
ωµ
xˆ
∂
∂x
E + e− jβ z + E − e+ jβ z
yˆ
∂
∂y
0
zˆ
∂
j ∂
+ − jβ z
− + jβ z 
= yˆ
E
e
+
E
e
)
 (
∂z
ωµ  ∂z
0
r − j β ( E + e− jβ z ) + ( E − e+ jβ z ) j β
− j β {( E + e − j β z ) − ( E − e + j β z )}
H=
( j ) yˆ =
( j ) yˆ
ωµ
=
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β {( E + e− jβ z ) − ( E − e + jβ z )}
ωµ
ωµ
yˆ =
1
η
Electromagnetic Field Theory by R. S. Kshetrimayum
[ E + e − j β z − E − e + j β z ] yˆ
3/19/2014
5.4 Plane waves in various media
η is the wave impedance of the plane wave
Ex
ωµ
µ
=
=
η=
β
ε
Hy
For free space,
ηo =
µo
= 120π = 377Ω
εo
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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
r
r
∇× E = − jωµ H ;
r
r
r
r
r
∇ × H = jω ε E + σ E = ( jω ε + σ ) E = jω ε eff E ;
ε eff (ω ) = ε +
σ
jσ
jσ 

=ε −
= ε 1 −

jω
ω
ωε


The effect of the conductivity has been absorbed in the
complex frequency dependent effective permittivity
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
r
r
r
2 r
2
2
⇒ ∇ E + ω µε eff (ω ) E = ∇ E + ( jγ ) E = 0
2
We can define a complex propagation constant
γ = jω µε eff (ω ) = α + j β
where α is the attenuation constant and β is the phase
constant
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
What is implication of complex wave vector?
The wave is exponentially decaying (see example 4.4).
The dispersion relation for a conductor (usually nonmagnetic) is
γ = jω µε eff (ω ) = jω µ0ε 0
ε eff (ω )
ω
= jω µ0ε 0 neff (ω ) = j neff (ω )
ε0
c
where neff is the complex refractive index
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
1-D wave equation for general lossy medium becomes
∂ 2 Ex
∂z 2
− γ 2 Ex = 0
whose solution is 1-D plane waves as follows
E x ( z ) = E + e −γz + E − e +γz = E + e −αz e − jβz + E − eαz e jβz
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
Putting the time dependence and taking real part, we get,
E x ( z, t ) = E + e −αz cos(ωt − βz ) + E − eαz cos(ωt + βz )
The magnetic field can be found out from Maxwell’s
equations as in the previous section
H y (z) =
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1
ηeff
[ E + e − γ z − E − eγ z ]
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
where useful expression for intrinsic impedance is
ηeff =
jωµ0
γ
=
jωµ0
jω µ0ε eff (ω )
=
µ0
ε eff (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
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Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
r r r*
E e e
+ −α z − j β z
ˆ
S = E × H = E e e x×

ηeff

+ −α z − j β z
*

e e
+ 2 −α z − j β z
ˆ
×
 y = E e e
*
η

eff
−α z + j β z
zˆ =
E
η
+ 2
*
e−2α z zˆ
eff
it is decaying in terms of square of an exponential function
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
49
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
For good dielectric,
Q σ << wε ∴ γ = jω µε ( 1 −
jσ
ωε
)
can be approximated using Taylor’s series expansion obtain α
and β as follows:
α=
σ
2
µ
ε
β = ω µε
For a good conductor, σ >> ωε
Therefore,
γ ≅ (1 + j )
50
ωµσ
2
Electromagnetic Field Theory by R. S. Kshetrimayum
⇒α = β =
wµσ
2
3/19/2014
5.4 Plane waves in various media
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
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
δ=
51
1
α
=
1
β
=
2
ωµσ
Electromagnetic Field Theory by R. S. Kshetrimayum
=
2
1
=
2π f µσ
π f µσ
3/19/2014
5.4 Plane waves in various media
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
Let us assume an EM wave which has x-component and
traveling along the z-axis
Then, it can be expressed as
E x ( z , t ) = E0 e −αz e − j ( βz −ωt )
52
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
5.4 Plane waves in various media
Taking the real part, we have,
E x ( z , t ) = E0 e −αz cos(ωt − βz )
Substituting the values of α and β for good conductors, we
have,
E x ( z , t ) = E0 e −
πfµσ z
(
cos ωt − πfµσ z
)
Now using the point form of Ohm’s law for conductors, we
can write
J x = σE x (z , t ) = σE0 e −
53
Electromagnetic Field Theory by R. S. Kshetrimayum
πfµσ z
(
cos ωt − πfµσ z
)
3/19/2014
5.4 Plane waves in various media
What is the phase velocity and wavelength inside a good
conductor?
vp =
54
ω
2π
= ωδ ; λ =
= 2πδ
β
β
Electromagnetic Field Theory by R. S. Kshetrimayum
3/19/2014
Electromagnetic
Waves
5.5 Summary
Plane waves
Plane waves in
various media
r r
)
)
)
) )
)
k • r = ( k x x + k y y + k z z ) • ( xx + yy + zz ) = k x x + k y y + k z z = con tan t
Polarization
Lossless
medium
r
ELP (z ) = (E0 x xˆ + E0 y yˆ )e − jβz
r
ERHCP = Eo ( xˆ − jyˆ )e− j β z


β = ω µε
r
E EP ( z ) = xˆ + Ae jφ yˆ e − jβz
(
Lossy
conducting
medium
)
vp =
Poynting vector
η=
ω
1
=
β
µε
ωµ
µ
=
β
ε
ε eff (ω ) = ε 1 −
55
2 ∂t
(H • H ) −
2 ∂t
α =β =
γ = jω µε eff (ω ) = α + j β δ =
ηeff =
r r
∂ 1
∂ 1 2
2
S
•
d
s
=
−
µ
H
dv
−
εE dv − ∫ σE 2 dv
∫S
∫
∫
∂t V 2
∂t V 2
V
r r
r
r ε ∂ r r
r r
µ ∂
∴∇ • ( E × H ) = −
jσ 

ωε 
Good
conductor
(E • E) − E • J
Electromagnetic Field Theory by R. S. Kshetrimayum
jωµ 0
γ
=
µ0
ε eff (ω )
vp =
1
α
=
1
β
Good
dielectric
ωµσ
α=
2
=
1
πfµσ
σ
2
µ
ε
β = ω µε
ω
2π
= ωδ ; λ =
= 2πδ
β
β
J x = σE x ( z , t ) = σE0 e −
πfµσ z
(
cos ωt − πfµσ z
Fig. 5.3 Plane waves in a nutshell
3/19/2014
)