[1] Maxwell's equations describe the fundamental interactions between electric and magnetic fields and how they propagate as electromagnetic waves. [2] In vacuum, Maxwell's equations simplify and predict that changing electric fields produce magnetic fields and vice versa, resulting in electromagnetic waves that propagate at the speed of light. [3] The document derives the wave equations for electric and magnetic fields in vacuum, showing that disturbances in these fields oscillate and regenerate themselves as they travel at the speed of light, representing electromagnetic waves.

1. ELECTROMAGNETIC WAVES

2. Divergence and curl of static Electric and Magnetic fields,
Farady Law and Maxwell’s correction for time dependent
fields, Maxwell’s equations in vacuum and media, em
waves and transverse nature of waves,
Assigned Part
References: Optics, A. Ghatak, Tata McGraw-Hill (2011).
Introduction to Electrodynamics, D. J. Griffith, Pearson, 2007

3. GENERAL CONCEPT
Divergence and Curl of vector fields:
sources of those fields
If we know a field F , we can calculate directly its source
Field divergence(if source is a scalar) Curl (if source is a vector)
OR
If one knows the source, say a scalar function D, or a vector
function C for which C = 0, one can workout the field by solving
the corresponding differential equation:

4. Geometrical Interpretation
The divergence div(u) = .u is positive when the vectors u are spreading out,
and negative when they are coming together.
diverging: .u > 0 converging: .u < 0
In 2D, curl(u) > 0 in regions where the field is curling anticlockwise, curl(u)<0,
curling clockwise. Absolute value of curl(u) determined strength of curling.
curl(u) > 0, smaller curl(u) < 0, larger
Curl is simply the circulation per unit area, circulation density, or rate of rotation
(amount of twisting at a single point).

5. The electric field of a source(point charge or charge
distribution) can graphically represented by field lines.
The length of the arrow represents the magnitude of electric field.
Electric field lines

6. The density of field lines represents proportionally the magnitude
of the electric field.
Continued…

10. The above equation can be read as follows : the source of electric
field is an electric charge distribution, in particular we can have an
electric charge monopole to produce the electric field.

11. • Looking at the radially directed electric field lines produced by the
point charge, we can deduce that curl of electric field is zero.
small calculation, shows   E = 0 HW# check this relation ?

12. Divergence and Curl of Magnetic field
BAR MAGNET
CURRENT CARRYING SOLINOID
ELECTRIC FIELD
MAGNETIC FIELD
Divergence:  . B = 0
In magnetostatics, it can be proved that the curl of
magnetic field B is given by
A magnetic field has no divergence, which is a
mathematical statement that there are no magnetic
monopoles
This means that there are no point sources of
magnetic field lines, instead the magnetic fields
form closed loops round conductors where current
flows.
Curl:   B = 0j
j : current density
Now we can proceed for Maxwell equations:

13. We can learn how a vast range of physical phenomena follow
from Maxwell’s equations...

14. Maxwell’s equations
Maxwell's equations are a set of partial differential equations that, together
with the Lorentz force law, form the foundation of classical electrodynamics,
classical optics, and electric circuits. These fields in turn underlie modern
electrical and communications technologies.
Importance:
[1] Describe how electric charge and electric currents act as
sources for the electric and magnetic fields.
[2] Describe how a time varying electric field generates a time
varying magnetic field and vice versa.
[3] Of the 4 equations, two of them: Gauss law and gauss law for
magnetism
[4] other two describe how the field “circulate” around their
respective sources:
The magnetic field circulates around electric currents and time
varying electric field in Ampere's law with Maxwell correction,
while the electric field circulates around time varying magnetic
fields in Faradays law.

15. Maxwell's Equations with sources in free space
Equation-1: : Gauss’s Law for the Electric Field
The flux of the E Field through a closed surface is due to the charge density
contained inside
 
VS
E dVdSEFlux 4.
Recall the divergence theorem for a vector A
(Closed surface integral of A is equal to the volume integral of the divergence of A)
 
S V
dVAdSA ..
and apply it to Gauss' Law for E (The surface integral of E is equal to the
volume integral of the divergence of E which is equal to
  
VS V
dVdVEdSE 4..
Since the integrals are equal for any volume the integrands are equal too, giving us
the differential form of the Law:
 
VV
dVdVE 4.
What does it mean?
The flux of the E field though a closed surface is due to the charge density
contained inside Electric charges produce electric fields
4.  E ..........[1]

16. Equation-2: Gauss’s Law for the Magnetic Field
The B field is a dipole field so no matter how small the volume is you will always
find equal numbers of north and south poles. So if you integrate over a closed
surface you’ll always get a net magnetic flux of zero. In integral form it is:
 
S
B dSBFlux 0.
Use the following Divergence theorem (Surface integral of A is equal to the volume
integral of the divergence of A)
 
VS
dVAdSA ..
Apply it to Gauss Law for B (Closed surface integral of B is equal to the volume
integral of the divergence of B which is equal to zero)
0..   VS
dVBdSB
Since the integrals are equal for any volume the integrands are equal too, giving us
the differential form of the Law: 0. V
dVB
What does it mean?
The flux of the B field through a closed surface is zero and no matter how much
we wish magnetic monopoles do not exist
0.  B ................[2]

17.  
S
B dSBFlux 0.



S
dSB
dt
d
cdt
d
c
emf .
11
 
C
dlEemf 0.
 
SC
SdB
dt
d
c
dlE .
1
.
Equation-3: Faraday’s Law
Since del dot B is exactly zero, we have an interesting result. If we don't close the
surface integral we get a magnetic flux. And a magnetic flux that changes in time
produces an emf, that is, a non conservative circulating E field with a nonzero
closed line integral
Recall Stoke's Theorem for a vector A
(Closed line integral of A is equal to the open surface integral of the curl of A)
SdAdlA
SC
..  
Apply it to the integral form of the Law : SdB
dt
d
c
SdAdlA
SSC
  .
1
..
Since the integrals are equal for any surface, the integrands are equal too, giving
us the differential form of the Law:
t
B
c
E



1 ..................[3]

18. Equation-4: Ampere's Law with conservation
The line integral of the B field around a closed path is equal to the surface integral
of the current density flow through a surface bound by the path. In integral form
 
C S
SdJ
c
ldB .
4
.

Use Stoke's Theorem (Line integral of A is equal to the surface integral of the curl of
A)
  
C S
SdAldA ..
And apply it to the integral form of the Law:   
SC S
SdJ
c
SdBldB .
4
..

Since the integrals are equal for any surface, the integrands are equal too, giving
us the differential form of the Law:
J
c
B
4

But that's not all! If you take the Divergence of Ampere's Law J
c
B
4
.. 
tc
J
c
B



 44
.0.
Because
t
J




.
That is net outflow of current is equal to the rate at which
the charges are lost. That's charge conservation

19. But we know that a changing E field produces a B field and if you take the partial
time derivative of Gauss' electric law you get a current term.
4.
t
E
t 




or tt
E







4.
t
E


Let us define the ‘Displacement Current’:
Put it into Ampere's equation so that it obeys charge conservation
What does it mean?
The curl of the B field is due to the current flow and a changing electric field
t
E
c
J
c
B



14 .................[4]
4.  E
0.  B
t
B
c
E



1
t
E
c
J
c
B



14
Maxwell's Equations with sources in free space

20. Equation-I: The electric field diverges in regions where there is
positive charge and converges in regions where there is
negative charge.
Equation-II: The magnetic field never diverges or converges.
Equation-III: Changing magnetic fields cause the electric field
to curl.
Equation-IV: Currents cause the magnetic field to curl.
Changing electric fields also cause the magnetic field to curl,
but the effect is usually much weaker, because 0 is small.
Maxwell-equations means

21. P, the polarization vector, is the electric dipole density induced by the external
field, E.
P weakens E so we define the Displacement D which is the field due only to
charges that are free
PED 4
M, the magnetization,is the magnetic dipole density induced by the external field, B.
M strengthens B and so we define the H which is the field due only to the currents
that are free.
MBH 4
Maxwell's Equations in media
The free space equations are not valid in the presence of matter, in media because
the E and B fields produce polarization (P) and magnetization (M) effects in the
bound charges of the material.
Then Gauss Law becomes
4.  D t
D
c
J
c
H



14
Ampere's law becomes
Then the Maxwell’s equation in Media become
4.  D
t
D
c
J
c
H



14
0.  B
t
B
c
E



1

22. Since there’s no sources set the ρ=0 and J=0
Then the Maxwell's equation in vacuum become
0.  E
0.  B
t
B
c
E



1
t
E
c
B



1
Maxwell's Equations in Vacuum
What happens when there are no currents, no charges within? Then
everything simplifies in this special case we have Maxwell's equations in
empty space!

23. The Wave Equations(electromagnetic)
Start with Maxwell's Equation in a Vacuum
0.  E 0.  B t
B
c
E



1
t
E
c
B



1
and take the curl of Faraday's Law.
t
B
c
E



1
Using the following simplification
AAA 2
).()( 
Then the curl on Faraday’s law could be written as
But we know 0.  E
then EE 2
)( 
EEE 2
).()( 
But again we know
t
B
c
E



1
Substituting in the equation we get 2
2
11
t
E
c
B
tc 





Put it all together and we derive a wave equation for the E field:
We get 2
2
2 1
t
E
c
E


 or 0
1
2
2
2




t
E
c
E
HW#
and
t
E
c
B



1

24. Similarly for magnetic field we get
0
1
2
2
2




t
B
c
B
A changing E produces B
A changing B produces E
Electromagnetic Fields are oscillating and
regenerating at the speed of light.
Electromagnetic waves travelling at the
speed of light.

25. Electromagnetic Waves
Vacuum…
Changing magnetic flux  Electric field : KFL
Changing electric flux  Magnetic field : AM
Alternating magnetic or electric field
 ELECTROMAGNETIC WAVE
++ -Speed in vacuum:
smc /103
1 8
00



26. Propagation in vacuum
Not mechanical……..Disturbance in EM field
aether properties….unrealistic and unobserved
Plane wave: E  B
Can be very high frequency.
*Limited only by rate of change in the field.
f = c
Direction and wave motion is E  B….. both

27.  It consists of mutually perpendicular and oscillating electric and
magnetic fields. The fields always vary sinusoidally. Moreover, the
fields vary with the same frequency and in phase (in step) with each
other.
 The wave is a transverse wave, both electric and magnetic fields are
oscillating perpendicular to the direction in which the wave
travels. The cross product E x B always gives the direction in which
the wave travels.
 Electromagnetic waves can travel through a vacuum or a material
substance.
 All electromagnetic waves move through a vacuum at the same
speed, and the symbol c is used to denote its value. This speed is
called the speed of light in a vacuum and is: c = 299792458 m/s
 ;sin tkxEE m   tkxBB m  sin
 The magnitudes of the fields at every instant and at any point are
related by
c
B
E
 (amplitude ratio)

28. Poynting Vector
 The rate of energy transport per unit area in EM wave is
described by a vector, called the Poynting vector
• The direction of the Poynting vector of an electromagnetic
wave at any point gives the wave's direction of travel and the
direction of energy transport at that point.
• The magnitude of S is
2
00
11
E
c
EBS


BES 
0
1

The electromagnetic waves carry energy with them and as they
propagate through space they can transfer energy to objects
placed in their path.
2
0
B
c
S

or

29. The root-mean-square value of the electric field, as
• The energy associated with the electric field exactly equals to the
energy associated with the magnetic field.
Intensity of EM Wave
2
1
)](sin[
1
][
1 2
0
22
0
2
0
m
avgmavgavg
E
c
tkxE
c
E
c
SI




2
0
1
rmsE
c
I


2
m
rms
E
E 
0
2
2
00
0
2
0
2
0
2
1
2
1
)(
2
1
2
1


B
BcBEuE 
BE uu 
The total instantaneous energy density: u = uE+ uB = oE2 = B2/o
What is of greater interest for a sinusoidal plane EM wave is the
time average of S over one or more cycles, which is called the
wave intensity I

30. ASSIGNMENT-1
1. Show that if a beam of unpolarised light strikes a polarised sheet, the
intensity of the transmitted light is one half that of the incident light.
2. A linearly polarised beam of intensity 800 W/m2 is incident on an analyzer.
Its electric field vector is parallel to the transmission axis of the analyser. Find
the intensity of the beam transmitted by the analyser.
3. What is the Brewster’s angle for a piece of glass(n=1.56) submerged in
water?
4. Given the intensity of an electromagnetic wave I = 1380 W/m2 . Find the
electric as well as magnetic field component of the electromagnetic waves at
any instant of time.
5. Check the unit and dimension of Eo/Ho.
Due Date: 20-09-2013
AAA 2
).()( 
6. Write 4 Maxwell’s equations both in differential and integral form.
7. Prove using vector identity that . the symbols
are their usual meaning.