5. Plane Electromagnetic Waves - iitg.ac.in · 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

Dr. Rakhesh Singh Kshetrimayum

5. Plane Electromagnetic Waves

Dr. Rakhesh Singh Kshetrimayum

3/19/20141 Electromagnetic Field Theory by R. S. Kshetrimayum

5.1 Introduction Electromagnetic

Waves

Plane waves Poyntingvector

Plane waves in various media

3/19/2014Electromagnetic Field Theory by R. S. Kshetrimayum2

Polarization Lossless medium

Good conductor

Fig. 5.1 Plane Waves

Lossyconducting medium

Good dielectric

5.2 Plane waves5.2.1 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?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 Field Theory by R. S. Kshetrimayum

( ) ( )0

,j k r t

F r t F eω• −

=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

kr

rr

Fr


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

0Fr

0Fr

0Er

0Hr

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 wavesProperties 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 H

2η0

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 waves5.2.2 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


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 wavesTypes 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 −− +=+= ˆˆˆˆ00

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 yyxxˆcosˆcos, 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 eyExEeyExEzEββ −− +=+= ˆˆˆˆ

00

r

5.2 Plane waves


The amplitude of the electric field vector is given by

( ) ( ) ( )yztExztEtzE yxLPˆcosˆcos, 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 tany y

LP

x x

E E

E Eθ − −

= =

( ) ( ) ( ) ( )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 oxLPˆcos, 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 oxz

LPˆcos,

0ω=

=

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 wavesCircularly polarized (CP) wave:

Now consider the case Ex = j Ey = Eo, where Eo is real so that

0 20 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 eβ−= −

r

ˆ ˆ( , ) [ cos( ) cos( )]2

RHCP oE z t E x t z y t z

πω 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 1sintantan tan

cosRHCP

tt t

t

ωθ 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 wavesElliptically 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,


( ) ( ) zjj

EP eyAexzEβφ −+= ˆˆ

r

( ) ( ) ( )yztAxzttzE EPˆcosˆcos, 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

zE t x A t yω ω φ

== + +

r

5.2 Plane waves

( ) ( ) ( )

( )( ) ( )

00

00

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


( ) ( ) ( )

( ) ( ) ( )

00

00

ˆ ˆ( ) 1, ; cos sin2

ˆ ˆ( ) 1, ; cos sin2

z

z

c A E E t x y t LHCP

d A E E t x y t RHCP

πφ ω ω

πφ ω ω

=

=

= = = −

= = − = +

r

r

5.2 Plane waves

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

00

00

0

ˆ ˆ( ) 3, ; cos 3sin2

ˆ ˆ( ) 0.5, ; cos 0.5sin2

ˆ ˆ( ) 1, ; cos cos4 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

πφ ω ω

πφ ω ω

π πφ ω ω

=

=

= = = −

= = − = +

= = = + +

r

r

r


( ) ( )

( ) ( )

00

00

ˆ ˆ( ) 1, ; cos cos4 4

ˆ ˆ( ) 1, 3 ; cos cos 34 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 πφ 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

( )*1Re

2avgS E H= ×r r r


Let us try to derive the point form of Poynting theorem from two Maxwell’s curl equations

t

HE

∂

∂−=×∇

rr

µ Jt

EH

rr

r+

∂

∂=×∇ ε


From vector analysis,

We can further simplify

)()()()()( Jt

EE

t

HHHEEHHE

rr

rr

rrrrrrr+

∂

∂•−

∂

∂−•=×∇•−×∇•=×•∇ εµ

( )AAtt

AA

rrr

rQ •

∂

∂=

∂

∂•

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 Jt t

µ ε∂ ∂∴∇ • × = − • − • − •

∂ ∂

r r r r r r r r

( )AAtt

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

dvJEdvEEt

dvHHt

rrrrrr)(

2)(

2

εµ( ) ( ) ∫∫∫ •=•×=×•∇SSV

sdSsdHEdvHErrrrrrr


We can interchange the volume integral and partial derivative w.r.t. time

∫∫∫∫ −∂

∂−

∂

∂−=•

VVVS

dvEdvEt

dvHt

sdS 222

2

1

2

1σεµ

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 H

2η0


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

00

ˆˆˆˆ

ˆ1ˆ

ηηη

ηηββ

zEEEzzEEEEz

EzEeEz

eEHESzjzj

rrrrrrr

rrr

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

0ˆˆ

Re1

Re1 zEzE

HESavg

rrrrr

=

=×= ∗


o Stored Electric Energy:

o Stored Magnetic Energy:

( )00 2

Re2

Re2 ηη

HESavg =

=×=

2

0

1

2e

w Eε=

2

2 2 20

0 0 0 02

0 0

1 1 1 1

2 2 2 2m e

Ew H E E w

εµ 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 +=2Q


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.,

0E 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

zβ


where E+ and E- are arbitrary constants

ej z j z

xE E E e

β 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 bv

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 /1031 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


HjErr

ωµ−=×∇

( )

ˆ ˆ ˆ

ˆ ej z j z

x y z

E j E j jH y E E e

β β+ − − +∇× ∇× ∂ ∂ ∂ ∂ = − = = = +

r rr


( )ˆ e

e 0 0j z j z

H y E E ej x y z z

E E eβ β

ωµ ωµ ωµ ωµ+ − − +

= − = = = + ∂ ∂ ∂ ∂

+

( )

( )

( ) ( )( ) ( )ˆ ˆ

( ) ( ) 1ˆ 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 jH j y j y

E e E ey 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

ωµη


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=== 377120πε

µη

o

oo


;E j Hωµ∇× = −r r

( ) ;effH j E E j E j Eω ε σ ω ε σ ω ε∇ × = + = + =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 σε ω ε ε ε

ω ω ωε

= + = − = −


We can define a complex propagation constant

( ) ( )22 2 2

0eff

E 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 ωγ 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 Ez

Eγ


zjzzjzzzx 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(),( zteEzteEtzEzz

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 µη

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 lossyconducting 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 zz j z z j z z

eff

EE e e e eS E H E e e x y E e e z e z

α β α βα 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 using Taylor’s series expansion obtain αand β as follows:

( 1 )j

w jσ

σ ε γ ω µεωε

<< ∴ = −Q

ε

µσα

2= µεωβ =


For a good conductor,

Therefore,

εα

2= µεωβ =

ωεσ >>

22)1(

µσβα

ωµσγ

wj ==⇒+≅


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

1δ

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 eeEtzEωβα −−−= 0,


Taking the real part, we have,

Substituting the values of α and β for good conductors, we have,

( ) ( )zteEtzEz

x βωα −= −cos, 0


have,

Now using the point form of Ohm’s law for conductors, we can write

( ) ( )zfteEtzEzf

x µσπωµσπ −= −cos,

0

( ) ( )zfteEtzEJzf

xx µσπωσσ µσπ −== −cos,

0


What is the phase velocity and wavelength inside a good conductor?

2; 2

pv

ω πωδ λ πδ

β β= = = =


5.5 SummaryElectromagnetic

Waves

Plane waves

PolarizationLossless

Good conductor

Plane waves in various media

Lossyconducting

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 eβ−= −

r

∫∫∫∫ −∂

∂−

∂

∂−=•

VVVS

dvEdvEt

dvHt

sdS 222

2

1

2

1σεµ

rr

( ) ( ) ( )2 2

E H H H E E E Jt t

µ ε∂ ∂∴∇ • × = − • − • − •

∂ ∂

r r r r r r r r

µεωβ =

µεβ

ω 1==pv

ε

µ

β

ωµη ==

( )

−=

ωε

σεωε

jeff 1


( )ωε

µ

γ

ωµη

eff

eff

j 00 ==

ε

µσα

2=

µεωβ =µσπβα

δf

111===

2

ωµσβα ==

2; 2

pv

ω πωδ λ πδ

β β= = = =

( ) ( )zfteEtzEJzf

xx µσπωσσ µσπ −== −cos, 0

( ) ( ) zj

yxLP eyExEzEβ−+= ˆˆ

00

r

( ) ( ) zjj

EP eyAexzEβφ −+= ˆˆ

r

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5. Plane Electromagnetic Waves - iitg.ac.in · 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