1.
M A N M O H A N D A S H ,
P H Y S I C I S T , T E A C H E R
Physics for ‘Engineers and Physicists’
Lecture - 3
Electromagnetic Waves

2.
I remember its close to 2 decades ago,
when I saw the Maxwell's Equations and
see those charge and currents and think
to myself; if these things vanish why still
we have the fields, I mean we read in the
text books the sources of charges is what
produces the electric fields and the
currents are what produces magnetic
fields, why the fields survive when the
charges vanish.
I was still thinking like the +2
(PU/American Highschool Senior)
student that I was, because that's when I
studied the fields being produced by the
charges and currents. Somewhere down
the line, when I became well versed with
what these equations are actually doing, I
understood; the fields are produced by
charges and currents, that are not
necessarily explicitly seen in these
equations, but the charges and currents
that interact with these fields are
explicitly placed in the equations.
Its like we are produced by our parents
who are not visible much in our lives
although they impact us, but its our
girlfriends and boyfriends who impact us
explicitly, trying to pull the strings here
and there !
It was just a passe analogy. But Physics is
often fun if you imagine how they
correspond to our day to day moorings.
Lets discover what
electromagnetic phenomena
are entailed by the Maxwell’s
equations.

3.
“A concise course of important results”
Lectures delivered around 12.Nov.2009 +
further content developments this week; 28
Aug - 02 Sept 2015 !

4.
Electromagnetic Waves
Electromagnetic Waves are 3 dimensional propagation of
vibrations of electric and magnetic fields. In the last two
lectures we discussed what are vector fields. Electric fields
and magnetic fields are vector fields.
>> Lecture-1 << and >> Lecture-2 <<
We discussed the analytical properties of vector and
scalar fields in detail in those lectures. Have a look,
they are linked below.

5.
Transverse waves
Electromagnetic Waves are transverse in nature. Waves
are basically two types transverse and longitudinal.
Note; vibrations are also called disturbances, undulation, oscillation, ripples
and wiggles in various usage of waves. Electromagnetic waves are
transverse undulations of electric and magnetic fields.
Transverse Waves are so called because in this type the
direction of propagation of wave is perpendicular or
transverse to the direction of vibration.
The example of longitudinal waves are acoustic waves
(sound). They are longitudinal because the direction of
propagation of wave is along the direction of vibration.

6.
Transverse nature of electromagnetic waves. The vector field
B vibrates along y-direction whereas the vector field E
vibrates along x-direction. The wave cruises along the z-
direction. Thus the E-Mwave is vector in nature in addition
to being transverse. This results in photons being vector
particle or spin-1 in nature, an advance concept we won’t
discuss here.

7.
Various types of em-waves
Electromagnetic Waves are a set of phenomena broadly
categorized as “Gamma rays, X-rays, Ultraviolet Rays,
Visible light, Infra-red Rays, Microwaves and Radio waves.

8.
Spectrum of Electromagnetic Waves
Here is the quantitative spectrum of the electromagnetic
waves.

9.
E-M Waves from Maxwell’s Eqn
Electromagnetic Waves are comprehensively described by
Maxwell’s Equations, these equations we discussed
previously in Lecture-2 !
Lets rewrite the Maxwell’s Equations in a medium in
presence of charges and currents. We will use the same to
derive whats called wave equation (for EM-phenomena).
We saw in Lecture-2 ! D and H are defined with mu, eps.
medium
B
H





10.
E-M Wave equation in free space from Maxwell’s Eqn
In Lecture-2 we saw that for free space or vacuum
Assuming no charges or currents, i.e.
lets write the Maxwell’s Equations again, this time in
freespace, free from sources of charges and currents.
0,0  j


0)4(
0)3(
0,0)2,1(
00 








t
E
B
t
B
E
BE






Lets take curl of the 3rd
equation of this set and
replace the curl of magnetic
field B from equation 4, into
the curl of 3rd.
Maxwell’s Equations,
freespace

11.
E-M Wave equation in free space, in Electric Field E.
In Lecture-1 we saw that !
0)4(
0)3(
,0,0)2,1(
00 








t
E
B
t
B
E
BE






In taking curl of the 3rd equation we
get
AAA
 2
)( 
0
)(
)(
)( 2







t
B
EE
t
B
E




Maxwell’s Equations,
freespace
Equation 4th and 1st ;
t
E
B





00
This leads to
0 E

02
2
00
2




t
E
E


 Freespace Wave-Equation, in electric field E

12.
E-M Wave equation in free space, Magnetic Field B.
0)4(
0)3(
,0,0)2,1(
00 








t
E
B
t
B
E
BE






In taking curl of the 4th equation we get
0
)(
)(
)(
00
2
00 






t
E
BB
t
E
B





Maxwell’s Equations,
freespace
Equation 3rd and 2nd;
t
B
E





This leads to
0 B

02
2
00
2




t
B
B



Freespace Wave-Equation, in magnetic field B
Lets apply the exact same treatment to get the wave equation
in terms of magnetic field B, apply curl on 4th equation !

13.
E-M Wave equation in free space, general form.
In general any wave equation looks like
(1);
This gives speed of the E-M waves, as
given above >>
00
1

emv0
1
x
)1( 2
2
22
2






tv

The wave equations we obtained are vector equations which
we can cast into individual scalar equations for 3 scalar
components of the fields; x, y, z, for both fields; E and B.
0
0
0
2
z
2
00z
2
2
y
2
00y
2
2
x
2
00x
2












t
E
E
t
E
E
t
E
E



0,0,0 2
z
2
00z
2
2
y
2
00y
2
2
x
2
00x
2










t
B
B
t
B
B
t
B
B 
SISI 104,1085.8 7
0
12
0

 

14.
Wave equation in charge-free, non-conducting media
Its easy to see the forms in which they
pertain to charge-free but non-conducting
media, eg in (1), (2) ! Eps, mu; electro-
magnetic properties of the media.
In last slide we saw the wave equations for the freespace, by
freespace we meant charge-free and current-free space which
are essential conditions of vacuum.
0)2(
0)1(
2
2
2
2
2
2








t
B
B
t
E
E






1
)
1
(,
00
00







v
c
vcv
There is a drop in speed of light when it
enters any media from vacuum, speed of light
in vacuua is maximum and drops by a factor
eta known as refractive index of the media
and the refractive index is related to eps and
mu in freespace and in media.

15.
Wave eqn, charge-free, conducting media; Telegraph Eqn
Sigma is called the conductivity; of the given medium.
‘Charge-free and Conducting’ media is a special case
of the Maxwell’s Equations where we represent this
condition by the equations on right !
Ej




 ,0
Lets take curl of the 3rd equation and interchange curl and time differentiation !
From the steady-state and time-varying Maxwell’s Equations
given in slide 9, for any medium given by eps, mu and the
conditions above, we have
E
t
EB
t
B
EBE







 





 )4(,0)3(,0(2),0)1(

16.
Wave equation; The Telegraph Equations !
This leads to the telegraph equation in terms of E field, (5).
We apply the same treatment on equation (4) and obtain the
telegraph equation for the B field, (6).
With the step mentioned in last slide we obtain (3’);
E
t
E
B
t
B
EE





 





 )'4(,0
)(
-)()'3( 2
The terms on the right hand side of telegraph equations are dissipative terms.
t
B
t
B
B
t
E
t
E
E
















 2
2
2
2
2
2
)6(,)5( Telegraph Equations

17.
Vector Potential and Scalar Potential
In Lecture-1 we saw every vector field is associated with a
scalar field, the gradient of the scalar is the vector and we
called the scalar field as the potential. We saw that for many
vectors, there are potentials associated with each of them.
These vectors were always a space gradient (or any space
derivative) of the scalar potentials. Hence the name vector
field and (scalar) potential.
When it’s a scalar whose gradient creates a vector field it’s a
scalar potential and when it’s a vector whose curl (another
way to create space derivative of a vector) creates a vector
field the former vector is called a vector potential.

18.
Magnetic Vector Potential
In Lecture-1 we saw for every vector whose divergence
vanishes the vector is necessarily a curl of another vector and
for every vector that is a curl of another vector the divergence
of the former vector vanishes.
Since the divergence of the magnetic field is zero it implies
from the above, magnetic field vector B is curl of another
vector A, the magnetic vector potential.
FGthen0G
0GthenFGif
0;)(






if
FSince
potentialvectormagneticA
0)Cf(asCfAA
AB0;B






Since

19.
Electro-Magnetic Scalar Potential
We take the 3rd of Maxwell equations (check slide 15), eqn (1)
below, use the fact that the magnetic field is the curl of the
vector potential A we just defined, in last slide. Interchange
the order of the curl and the time derivative;
!,
t
-
-
t
0;)(
,0)
t
(,ABwith0,
t
E(1)
potentialscalar
A
E
A
E
fSince
A
E
B



























 We already saw
in Lecture-1 that
curl of any
gradient is
always 0.
Phi, the scalar
potential, is partly
electric and partly
magnetic potential.

20.
Gauge Transformations, Lorentz and Coulomb’s Gauge !
We saw that vector
potentials and scalar
potentials that we
just discussed are
arbitrary, that is,
there is not a single
one of them.
0;,
1
;..
.,;
),(,,;
2








AGaugeCoulomb
tc
AGaugeLorentzge
ConditionGaugecalleddconstraineAandGauged
trff
t
f
fAAtionsTransformaGauge






Gauge Conditions are therefore a
particularly chosen definitions of vector
or scalar potentials. The Gauge
Transformations are a change of the
definition of these potentials, so that the
E and B field stand unchanged, thus
Physics stays unique and not arbitrary;

21.
Wave Equation in terms of Scalar Potential
We saw that vector potential
and scalar potential, that we
just discussed are now unique,
given to a particular Gauge
Condition.
0
11
;
0
)(
;0)(
.0;,'x
2
2
2
2















tctc
AGaugeLorentzApply
t
A
or
t
A
E
ELawGaussEqnswellMaFreespace








Using this, we can write
the wave equation in
terms of the scalar
potential, instead of just
in E and B fields.
0
1
2
2




tc


wave equation
in terms of the
scalar potential

22.
Wave Equation in terms of Vector Potential
We saw that scalar potential
has its own wave equation just
as E and B fields did. Lets find
the wave equation for the
vector potential.
0][][
0
)(
)(
0)()(,0(1)
2
2
00
2
00
2
2
0000
2
0000
























t
A
A
t
A
t
A
t
AA
t
A
t
A
t
E
B















Using Freespace Ampere-
Maxwell Law (1), we can
write the wave equation
in terms of the vector
potential.
2
2
2
2 1
t
A
c
A





wave equation
in terms of the
vector potential
1st [term] is 0;
Lorentz Gauge

23.
Transverse nature of electromagnetic waves
In the beginning we claimed that electromagnetic waves are
transverse in nature, that is the propagation of the energy
and wave as such, occurs in a direction perpendicular to the
oscillations of the vector fields of E and B. Lets see this.
We see that the wave equations (1) are differential equations, 2nd order in space
and 2nd order in time. Plane waves (2) and (3) given above are their solutions.
Lets prove that E and B fields are perpendicular to each
other as well to the wave propagation direction; k vector.
Note that E is along e and B along b, the unit vectors.
)(
0
)(
02
2
00
2 ˆ),()3(,ˆ),()2(,0
),(
),()1( trkitrki
eBbtrBeEetrE
t
BE
BE 
 









24.
Transverse nature of electromagnetic waves
e and b are thus unit vectors, constant in space and time. k is
vector along which wave propagates, called wave vector and
its magnitude is called as wave number and gives wavelength
as well as momentum of the plane wave. Omega is the
angular frequency and gives the time period and frequency of
the wave, as well as the energy. e, b, k form a right hand trio.
E_zero and B_zero are the amplitudes or maximum value of
the fields. Below we see e and k are perpendicular.
0ˆ0))(iEˆ(or0)(ˆ
.ˆ;0ˆ),(ˆ)(ˆ0
)()(Use0;)ˆ(,0
)(
0
)(
0
)(
0
)(
0
)(
0






keekeeEe
constiseeeEeeEe
VAVAVAeEeE
trkitrki
trkitrki
trki









0ˆ ke

e and k are
perpendicular

25.
Homework
Homework; (1) prove the result we used in last slide;
)(
0
)(
0 )(iE)( trkitrki
ekeE  

 
Homework; (2) prove that b and k are also perpendicualar
to each other just like e and k.
0ˆ0  kbB

In the following slide we will prove from Maxwell’s relations, e, b, k are all
mutually perpendicular in a right handed way.

26.
Transverse nature of electromagnetic waves
Lets show that E, B and k are mutually perpendicular in a
right handed way, that is e, b and k are;
eeeEeEeeEe
AVAVVA
eBbeEe
t
B
E
rkititrkitrki
trkitrki
ˆ)()ˆ()ˆ(
)()(use
,0)ˆ(
t
)ˆ(0
0
)(
0
)(
0
)(
0
)(
0




















From homework (1) we saw, rkirki
ekie
  

See eg Lecture-1
This leads to; 0ˆ)ˆk(,)ˆk()ˆ( )(
0
)(
0
)(
0
)(
0   trkitrkitrkitrki
eBbieEeieEeieEe 

 
So we have the transverse condition; b
E
B
e ˆ)ˆk(
0
0


e, b and k are mutually RH-perpendicular
Lets prove this

27.
Transverse nature of electromagnetic waves
k has a magnitude equaling to product of angular frequency
and ratio of amplitude of magnetic field to that of electric
field, also note that for waves, angular frequency divided by
wave number is the speed of wave, hence the ratio of
amplitude of electric field to that of magnetic field is also the
speed of the wave;
Z_zero is called impedance of vacuum. It has dimension of
electric resistance.
0
0
0
00
0
0
0
0
0000
00
.,,
1
zc
H
E
cBEHBcAs 






E, B are in phase and their magnitude and directions are correlated.
0
0
0
0
0
0
,)()(k
B
E
cor
E
B
ck
E
B
 

28.
Electromagnetic Energy.
Electric Energy per unit volume or electric energy density
and Magnetic Energy per unit volume or magnetic energy
density are given by;
So the total energy density called the electromagnetic energy
density or energy per unit volume is given by;
22
2
1
2
1
,
2
1
2
1
HHBuEDEu BE  

)(
2
1
)(
2
1 22
HEHBDEuEM  


29.
Poynting vector and Poynting Theorem.
Poynting Vector measures the rate of energy flow of
electromagnetic waves per unit time, per unit area, normal to
the direction of wave propagation (SI unit, );

BE
HES

 

2
m
watt
Differential form (2) and integral form (1) of Poynting’s
Theorem. Intergral form is obtained by taking volume
integral and using Gauss Divergence Theorem. see Lecture-1
jE
t
u
S EM




)2( 



VV
EM
A
dVjEdV
t
u
AdS

)1(










VV
EM
A
VV
EM
A
dVjEdV
t
u
AdSLHSontheoremDivGaussUse
dVjEdV
t
u
dVSofIntegralVolumeTake


)1(;.
;)2(

30.
Proof of Poynting Theorem.
This proves the
differential form
of Poynting’s
Theorem from
Maxwell’s
equations.jE
t
u
S
jE
HE
t
HE
E
tt
E
E
t
D
ERHS
H
tt
H
H
t
B
HRHS
HEHEEHLHS
jE
t
D
E
t
B
HHEEHiiEiH
iij
t
D
Hi
t
B
E
EM






























































)
22
()(
)
2
(
)(
;
)
2
(
)(
;
)()()(;
)()()()(
)(),(
22
2
2



Maxwell’s equations.

31.
Significance of Poynting Theorem.
We already saw the significance of Poynting’s Theorem as the
quantitative relation depicting the flow of electro-magnetic
energy across regions in given time, this can be readily seen
as an equation of continuity;
If there are no sources of currents j, that the fields are
interacting with, it means no energy is dissipated that way,
we set j = 0 and we have the equation of continuity of energy
flow, which represents the conservation of electromagnetic
energy.
0,0 


 jwhen
t
u
S EM

equation of continuity

32.
Poynting vector and intensity of em-wave
Above we took the maximum value of Poynting’s vector, S =
EH. (cross product maximum) Realize that; the maximum
comes when the amplitudes are attained. For a plane wave E
is given as a time varying sine function, whose integral
average is ½ and thus given as rms value of the field.
2
2
2
0
2
0
22
0
0
2
0
0
22
sin
,sin,
,
rms
rms
EcSI
Ec
Ec
c
E
c
tE
StEE
c
E
S
c
H
E
H
EEB
EHS














Intensity of electro-magnetic wave

33.
Electro-magnetic waves in conducting media
What happens when electromagnetic waves pass through
conducting media, such as metallic conductors and ionized
gases or electrolytes. For wave propagation along z–axis,
electric field decays with distance by an exponetial factor f. E
reduces to 1/e of its value after a distance of z which is called
skin-depth or penetration depth of conducting media.
)1(02
2
2







t
E
t
E
E



Electromagnetic-waves passing through conducting
media, the 3rd term in (1) gives dissipation. High frequency
e.m. waves cannot propagate through conducting media
like metals and ionized gases as seen in factor f.


 2
z,
z
2
 efatt

34.
Thank you
We discussed electromagnetic waves at the undergraduate
engineering level. Its also useful for the Physics ‘honors’
and ‘pass’ students.
This was a course I delivered to engineering first years, around
12th November 2009. But I have added contents, added the
diagrams and many explanations now; 28th Aug - 2nd Sept
2015.
More soon. Eg Quantum Mechanics and Applications (for
engineering degree and for grad and undergrad of Physics
Major students) Reach me @ g6pontiac@gmail.com