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Lecture 19 1
Lecture 12
Maxwells Equations and Electromagnetic Waves
Maxwells Equations - Waves DAlembert
equation - Electromagnetic Waves The e.m.
spectrum -Energy of Electromagnetic Waves -
Poynting vector - Refractive index - Spherical
waves - Electromagnetic Radiation as Particles
Lecture 19 2
t
EjB
t
BE
B
E
000
0
0
MAXWELLs EQUATIONS
Lecture 19 3
WAVES
Lecture 19 4
WAVE: a transfer of energy from one point to another via
a traveling disturbance
Lecture 19 5
WAVES on a string:
pulses and harmonic wave
Lecture 19 6
WAVE:1D
WAVE:2D
Lecture 19
3D cylindrical wave
3D spherical wave
Lecture 19 8
2
22
2
2
x
yv
t
y
)(),( vtxftxy )()(),( 21 vtxfvtxftxy
Differential equation of wave
motion (DAlembert):
General solution:
Lecture 19
Plane harmonic wave
)(),( 0 tkxsenEtxE
kv
P
k
2
2
Lecture 19 10
From Maxwells equations to e.m. waves
Lecture 19
t
EB
t
BE
B
E
00
0
0
IN VACUUM:
NO charges
NO current
Supposing that E and B are transeverse waves
orthogonal to the x-axis, which is the propagation direction
From Maxwells equations to e.m. waves
x
E
t
B
x
E
t
B
t
B
yz
zy
x 0
!!!!zero! are derivative z andy The
x
B
t
E
x
B
t
E
t
E
yz
zy
x
00
00
1
1
0
2
2
002
2
dt
Bd
dx
Bd yy
2
2
002
2
dt
Bd
dx
Bd zz
Lecture 19 12
2
2
002
2
dt
Ed
dx
Ed yy
2
2
002
2
dt
Ed
dx
Ed zz
2
2
002
2
dt
Bd
dx
Bd yy
2
2
002
2
dt
Bd
dx
Bd zz
From Maxwells equations to e.m. waves
2
2
00
2 dt
EdE
2
2
00
2 dt
BdB
Both magnetic and electric fields
satisfy DAlembert equation
Lecture 19
x
y
z
Campo
elettrico
Campo
magnetico
Electric
field
Magnetic
field
1. Ex=Bx=0 that is: e.m. waves are transverse-wave
2. The e.m. wave propagation velocity in vacuum is constant and given by:
3. Ez, Bz , Ey, By components satisfy the wave-equation, and harmonic
solutions can be given by:
4. The simple solution for E and B fields propagation in vacuum along x-axis
is to have E//y-axis and B//z-axis:
smcv
v /1031
1
1 8
00
200
00
plane in the lying axis, with )sin(),(
plane in the lying axis, with )sin(),(
00
00
(y,z)x-BtkxBtxB
(y,z)x-EtkxEtxE
FIRST IMPORTANT RESULTS:
axis// with )sin(),(
axis// with )sin(),(
00
00
zBtkxBtxB
yEtkxEtxE y
Lecture 19
2. And making the derivatives::
cuc
EBE
c
EB
BE
0
2
OTHER VERY IMPORTANT RESULTS:
c
EB Demonstration of
1. Assuming a plane wave with E//y and B//z:
t
B
x
E
t
BE
utkxBB
utkxEE
zy
zz
yy
)sin(
)sin(
0
0
00
00
000
0
0
)cos(
)cos(
cBE
cBk
BEBkE
tkxBt
B
tkxkEx
E
zz
yzy
zz
y
y
Lecture 19 15
E.M. WAVES PROPAGATION:
-Polarization
-Rays and wavefronts
Lecture 19 16
e.m. WAVES PROPAGATION
Lecture 19 17
POLARIZATION
Circular
polarization:the electric
field of the passing
wave does not change
strength but only
changes direction in a
rotary manner.
The shape traced out in a fixed
plane by the electric vector as such a
plane wave passes over it (a
Lissajous figure) is a description of
the polarization state.
Waves can oscillate with more than
one orientation.
Electromagnetic waves, such as
light exhibit polarization; sound waves
in a gas or liquid do not have
polarization because the medium
vibrates only along the direction in
which the waves are travelling.
Lecture 19 18
POLARIZING FILTERS
Lecture 19
PHASE of THE Harmonic WAVE: tkx
WAVE-FRONT: is composed by all the points where the field has the
same value. A wavefront is the locus of points having the same
phase: a line (or a curve) in 2-D, or a surface in 3-D.
PLANE WAVE
It propagates along the x axis, the electric field E(x,t)
has to have nul x components: Ex(x,t)=0
E has the same value in a plane that is orthogonal to the
x axis, that is: in the planes // (y,z)
Plane
wave-front
The simplest form of a wavefront is the plane wave, where the rays are parallel to one another. The light from this type of wave is referred to as collimated light. The plane wavefront is a good model for a surface-section of a very large spherical wavefront; for instance, sunlight strikes the earth with a spherical wavefront that has a radius of about 150 million kilometers (1 Angstrom). For many purposes, such a wavefront can be considered planar.
Lecture 19
BE
00
1
c
e.m. WAVES PROPAGATION
The wave velocity propagation in vacuum is:
The propagation direction is given by the direction of :
Wave front
Lecture 19
Huygens' principle provides a quick method to predict the propagation of a wavefront: for
example, a spherical wavefront will remain spherical as the energy of the wave is carried
away equally in all directions. Such directions of energy flow, which are always
perpendicular to the wavefront, are called rays creating multiple wavefronts.
HUYGENS PRINCIPLE: any point on a wave front of light may be regarded as the source
of secondary waves and that the surface that is tangent to the secondary waves can be
used to determine the future position of the wave front.
RAYS and WAVEFRONTS
Lecture 19 22
E.M. WAVES GENERATION
Lecture 19 23
Electromagnetic Waves generation
Electric and magnetic fields are coupled through Ampres and
Faradays laws
Once created they can continue to propagate without further input
Only accelerating charges will create electromagnetic waves
Lecture 19 24
Electromagnetic Waves generation
Lecture 19 25
Electromagnetic Waves
With the changing current restricted to a line, the fields
propagate with cylindrical symmetry outward from the current line.
The electric field is aligned parallel to the current and the
magnetic filed is aligned perpendicular to both the electric field
and to the direction of propagation. These are general features
of electromagnetic waves.
The current must change in time if it is to give rise to propagating
fields (as a steady current merely produces a static magnetic
field). We can translate this into a statement about the charges
whose flow gives rise to the current: The charges that give rise
to the propagating electric and magnetic fields must be
accelerating. Harmonically varying currents will give rise to
harmonically varying electric and magnetic fields.
Lecture 19 26
THE
ELECTRO-
MAGNETIC
SPECTRUM:
Lecture 19 27
Images taken of the Whirlpool galaxy recordiung radiation in
different frequency ranges (and a s consequensce different details
are revealed)
ELECTROMAGNETIC WAVES ARE REAL
Lecture 19 28
Lecture 19 29
The refractive index
Lecture 19 30
Electromagnetic Waves in a medium
Electromagnetic waves travel more slowly through a
medium by a factor n:
This defines n, the index of refraction.
Except for ferromagnets, the speed can be written:
r
rr
nc
vv
1
00
Or, using r instead of k:
Lecture 19 31
Energy and Poynting vector
Lecture 19 32
Energy and Poynting vector
The energy in an electromagnetic wave must be shared equally
between the electric and magnetic fields:
This energy is transported by the wave; u is the energy density [J/m3]; it varies with time because E and B fields depend on time, so
we consider a mean value
The Energy flux is given by the energy/(unit area * unit time):
Energy flux = cu
Lecture 19
BEcucEcES
cSES
tcSEtS
Ew
BwEw
c
em
BE
2
0
2
0
2
0
2
0
2
0
2
0
2
0
2
0
: vectorPoynting
:ugh power thro
:in ough energy thr
2
1 ;
2
1
Energy and Poynting vector
More precisely, the energy flowing through a given surface can be
expressed as the flux through the same surface of the vector:
This vector is called the Poynting vector.
Lecture 19
)cos(),(
)cos(),(
0
0
tkxBtxB
tkxEtxE
sm
J
2
1
:be willS of valueaverage the
)(cos),(
2
2
00
22
00
IcES
tkxcEtxS
mean
Energy and Poynting vector
The direction of the Poynting vector is the
direction in which the energy flow propagates
The module of the Poynting vector represents
the energy per unit time per unit area (assuming
a surface orthogonal to the propagation direction)
The flux of the Poynting vector through a
surface gives the power through the surface
The average value of the Poynting vector is the
intensity of the wave
An electromagnetic wave also carries
momentum, and can exert pressure (called
radiation pressure) on objects
Lecture 19 35
SPHERICAL WAVES
Lecture 19
SPHERICAL WAVES
sm
J
2
12
2
00 IcES
22
00 4)(2
1rrcEPower
The power has to be constant, for each value of the radius r, because the radiation
emitted must pass through any sphere that surrounds the source
The value of the amplitudes E0(r) and B0(r) will be E0/r, B0/r (they decrease as 1/r)
Lecture 19 37
Electromagnetic Radiation as
Particles
Lecture 19 38
In physics, a black body is an object
that absorbs all electromagnetic
radiation that falls onto it. No radiation
passes through it and none is reflected.
Despite the name, black bodies are not
actually black as they radiate energy as
well, since every object with non-zero
temperature radiates electromagnetic
waves, due to the motion of electrons
and protons. How much electromagnetic
radiation they give off just depends on
their temperature.
Black body radiation
Lecture 19 39
Prediction of classical mechanics
Ultraviolet disaster
Lecture 19 40
Black body radiation 1900
hvE Energy=Plank constant times frequency
Plancks crazy idea: the energy can not be divided into smaller and smaller amounts. It is
emitted in discrete packet, called quantum (Nobel Prize 1918)
h=6.6310-34Js
Electromagnetic radiation is not only a kind of wave, but also a kind of particle: wave-particle duality
Lecture 19 41
Electromagnetic Radiation as
Particles
Electromagnetic radiation can act as though it is
made of individual particles, called photons.
The energy of each individual photon depends on
the frequency of the radiation:
with
The photon also has
momentum:
Lecture 19 42
DIPOLE RADIATION
Lecture 19 43
DIPOLE RADIATION
Lecture 19 44
DIPOLE RADIATION
Lecture 19 45
DIPOLE RADIATION: angular distribution
Lecture 19 46
Summary
Using Maxwells equations to find an
equation for the electric field:
This is a wave equation, with solution:
And propagation speed:
Lecture 19 47
This is the speed of light, c!
The magnetic field obeys the same wave equation.
The amplitude of the magnetic field is related to the
amplitude of the electric field:
Also, the two fields are everywhere orthogonal:
Summary
Lecture 19 48
Electromagnetic waves are transverse the E
and B fields are perpendicular to the direction of
propagation
The E and B fields are in phase
Summary
Lecture 19 49
Summary
In the absence of free charges, electric and
magnetic fields obey a wave equation:
These waves propagate at the speed
of light:
Lecture 19 50
Summary
The waves:
Vary sinusoidally
Are transverse
Have related electric and magnetic field
intensities: E = cB
Have perpendicular electric and magnetic
fields
Carry energy and momentum