Download Chapter 34

Chapter 24
Electromagnetic Waves
Electromagnetic
Waves, Introduction




Electromagnetic (em) waves permeate
our environment
EM waves can propagate through a
vacuum
Much of the behavior of mechanical
wave models is similar for em waves
Maxwell’s equations form the basis of
all electromagnetic phenomena
Conduction Current


A conduction current is carried by charged
particles in a wire
The magnetic field associated with this
current can be calculated using Ampère’s
Law:
 B  ds   I
o

The line integral is over any closed path through
which the conduction current passes
Conduction Current, cont.



Ampère’s Law in this
form is valid only if the
conduction current is
continuous in space
In the example, the
conduction current
passes through only S1
This leads to a
contradiction in
Ampère’s Law which
needs to be resolved
Displacement Current

Maxwell proposed the resolution to the
previous problem by introducing an
additional term


This term is the displacement current
The displacement current is defined as
d E
Id   o
dt
Displacement Current, cont.

The changing electric field may be
considered as equivalent to a current



For example, between the plates of a capacitor
This current can be considered as the
continuation of the conduction current in a
wire
This term is added to the current term in
Ampère’s Law
Ampère’s Law, General

The general form of Ampère’s Law is
also called the Ampère-Maxwell Law
and states:
d E
 B  ds  o (I  Id )  oI  oo dt

Magnetic fields are produced by both
conduction currents and changing
electric fields
Ampère’s Law,
General – Example

The electric flux through
S2 is EA





S2 is the gray circle
A is the area of the
capacitor plates
E is the electric field
between the plates
If q is the charge on the
plates, then Id = dq/dt
This is equal to the
conduction current
through S1
James Clerk Maxwell






1831 – 1879
Developed the
electromagnetic theory
of light
Developed the kinetic
theory of gases
Explained the nature of
color vision
Explained the nature of
Saturn’s rings
Died of cancer
Maxwell’s Equations,
Introduction



In 1865, James Clerk Maxwell provided a
mathematical theory that showed a close
relationship between all electric and magnetic
phenomena
Maxwell’s equations also predicted the
existence of electromagnetic waves that
propagate through space
Einstein showed these equations are in
agreement with the special theory of relativity
Maxwell’s Equations

In his unified theory of
electromagnetism, Maxwell showed that
electromagnetic waves are a natural
consequence of the fundamental laws
expressed in these four equations:
q
 E  dA  
o
d B
 E  ds   dt
 B  dA  0
d E
 B  ds  oI  o o dt
Maxwell’s Equations, Details

The equations, as shown, are for free space


No dielectric or magnetic material is present
The equations have been seen in detail in
earlier chapters:




Gauss’ Law (electric flux)
Gauss’ Law for magnetism
Faraday’s Law of induction
Ampère’s Law, General form
Lorentz Force

Once the electric and magnetic fields
are known at some point in space, the
force of those fields on a particle of
charge q can be calculated:
F  qE  qv  B

The force is called the Lorentz force
Electromagnetic Waves


In empty space, q = 0 and I = 0
Maxwell predicted the existence of
electromagnetic waves


The electromagnetic waves consist of oscillating
electric and magnetic fields
The changing fields induce each other which
maintains the propagation of the wave


A changing electric field induces a magnetic field
A changing magnetic field induces an electric field
Plane em Waves


We will assume that the
vectors for the electric and
magnetic fields in an em
wave have a specific spacetime behavior that is
consistent with Maxwell’s
equations
Assume an em wave that
travels in the x direction with
the electric field in the y
direction and the magnetic
field in the z direction
Plane em Waves, cont



The x-direction is the direction of propagation
Waves in which the electric and magnetic
fields are restricted to being parallel to a pair
of perpendicular axes are said to be linearly
polarized waves
We also assume that at any point in space,
the magnitudes E and B of the fields depend
upon x and t only
Rays




A ray is a line along which the wave travels
All the rays for the type of linearly polarized
waves that have been discussed are parallel
The collection of waves is called a plane
wave
A surface connecting points of equal phase
on all waves, called the wave front, is a
geometric plane
Properties of EM Waves


The solutions of Maxwell’s are wave-like, with
both E and B satisfying a wave equation
Electromagnetic waves travel at the speed of
light
1
c
oo

This comes from the solution of Maxwell’s
equations
Properties of em Waves, 2

The components of the electric and
magnetic fields of plane electromagnetic
waves are perpendicular to each other
and perpendicular to the direction of
propagation

This can be summarized by saying that
electromagnetic waves are transverse
waves
Properties of em Waves, 3

The magnitudes of the fields in empty
space are related by the expression
c E


B
This also comes from the solution of the
partial differentials obtained from Maxwell’s
Equations
Electromagnetic waves obey the
superposition principle
Derivation of Speed –
Some Details

From Maxwell’s equations applied to empty
space, the following partial derivatives can be
found:
 2E
 2E
 o  o 2
2
x
t


and
 2B
 2B
 o  o 2
2
x
t
These are in the form of a general wave
equation, with v  c  1 o o
Substituting the values for o and o gives
c = 2.99792 x 108 m/s
E to B Ratio – Some Details

The simplest solution to the partial differential
equations is a sinusoidal wave:



The angular wave number is k = 2 p / l


E = Emax cos (kx – wt)
B = Bmax cos (kx – wt)
l is the wavelength
The angular frequency is w = 2 p ƒ

ƒ is the wave frequency
E to B Ratio – Details, cont

The speed of the electromagnetic
wave is
w 2p ƒ

 lƒ  c
k 2p l

Taking partial derivations also gives
Emax w E
Bmax

k

B
c
em Wave Representation


This is a pictorial
representation, at
one instant, of a
sinusoidal, linearly
polarized plane
wave moving in the
x direction
E and B vary
sinusoidally with x
Doppler Effect for Light

Light exhibits a Doppler effect


Remember, the Doppler effect is an
apparent change in frequency due to the
motion of an observer or the source
Since there is no medium required for
light waves, only the relative speed, v,
between the source and the observer
can be identified
Doppler Effect, cont.

The equation also depends on the laws of
relativity
c v
ƒ'  ƒ
c v




v is the relative speed between the source and the
observer
c is the speed of light
ƒ’ is the apparent frequency of the light seen by
the observer
ƒ is the frequency emitted by the source
Doppler Effect, final

For galaxies receding from the Earth v
is entered as a negative number



Therefore, ƒ’<ƒ and the apparent
wavelength, l’, is greater than the actual
wavelength
The light is shifted toward the red end of
the spectrum
This is what is observed in the red shift
Heinrich Rudolf Hertz


1857 – 1894
Greatest discovery
was radio waves



1887
Showed the radio
waves obeyed wave
phenomena
Died of blood
poisoning
Hertz’s Experiment


An induction coil is
connected to a
transmitter
The transmitter
consists of two
spherical electrodes
separated by a
narrow gap
Hertz’s Experiment, cont




The coil provides short voltage surges to the
electrodes
As the air in the gap is ionized, it becomes a
better conductor
The discharge between the electrodes
exhibits an oscillatory behavior at a very high
frequency
From a circuit viewpoint, this is equivalent to
an LC circuit
Hertz’s Experiment, final


Sparks were induced across the gap of the
receiving electrodes when the frequency of
the receiver was adjusted to match that of the
transmitter
In a series of other experiments, Hertz also
showed that the radiation generated by this
equipment exhibited wave properties


Interference, diffraction, reflection, refraction and
polarization
He also measured the speed of the radiation
Poynting Vector



Electromagnetic waves carry energy
As they propagate through space, they
can transfer that energy to objects in
their path
The rate of flow of energy in an em
wave is described by a vector called the
Poynting vector
Poynting Vector, cont

The Poynting Vector is
defined as
S


1
o
EB
Its direction is the
direction of propagation
This is time dependent


Its magnitude varies in
time
Its magnitude reaches a
maximum at the same
instant as the fields
Poynting Vector, final

The magnitude of the vector represents
the rate at which energy flows through a
unit surface area perpendicular to the
direction of the wave propagation


This is the power per unit area
The SI units of the Poynting vector are
J/s.m2 = W/m2
Intensity


The wave intensity, I, is the time
average of S (the Poynting vector) over
one or more cycles
When the average is taken, the time
average of cos2(kx-wt) = ½ is involved
2
2
Emax Bmax Emax
c Bmax
I  Savg 


2 o
2 o c
2 o
Energy Density




The energy density, u, is the energy per
unit volume
For the electric field, uE= ½ oE2
For the magnetic field, uB = ½ oB2
Since B = E/c and c  1  o  o
2
1
B
2
uB  uE   o E 
2
2 o
Energy Density, cont

The instantaneous energy density
associated with the magnetic field of an
em wave equals the instantaneous
energy density associated with the
electric field

In a given volume, the energy is shared
equally by the two fields
Energy Density, final

The total instantaneous energy density is the
sum of the energy densities associated with
each field


When this is averaged over one or more
cycles, the total average becomes


u =uE + uB = oE2 = B2 / o
uav = o (Eavg)2 = ½ oE2max = B2max / 2o
In terms of I, I = Savg = c uavg

The intensity of an em wave equals the average
energy density multiplied by the speed of light
Momentum



Electromagnetic waves transport momentum
as well as energy
As this momentum is absorbed by some
surface, pressure is exerted on the surface
Assuming the wave transports a total energy
U to the surface in a time interval Dt, the total
momentum is p = U / c for complete
absorption
Pressure and Momentum


Pressure, P, is defined as the force per unit
area
F 1 dp 1 dU dt
P 

A A dt c A
But the magnitude of the Poynting vector is
(dU/dt)/A and so P = S / c


For complete absorption
An absorbing surface for which all the incident
energy is absorbed is called a black body
Pressure and
Momentum, cont



For a perfectly reflecting surface,
p = 2 U / c and P = 2 S / c
For a surface with a reflectivity somewhere
between a perfect reflector and a perfect
absorber, the momentum delivered to the
surface will be somewhere in between U/c
and 2U/c
For direct sunlight, the radiation pressure is
about 5 x 10-6 N/m2
Determining
Radiation Pressure



This is an apparatus for
measuring radiation
pressure
In practice, the system
is contained in a high
vacuum
The pressure is
determined by the angle
through which the
horizontal connecting
rod rotates
Space Sailing


A space-sailing craft includes a very large sail
that reflects light
The motion of the spacecraft depends on the
pressure from light


From the force exerted on the sail by the reflection
of light from the sun
Studies concluded that sailing craft could
travel between planets in times similar to
those for traditional rockets
The Spectrum of EM Waves



Various types of electromagnetic waves
make up the em spectrum
There is no sharp division between one
kind of em wave and the next
All forms of the various types of
radiation are produced by the same
phenomenon – accelerating charges
The EM
Spectrum



Note the overlap
between types of
waves
Visible light is a
small portion of
the spectrum
Types are
distinguished by
frequency or
wavelength
Notes on The EM Spectrum

Radio Waves



Wavelengths of more than 104 m to about 0.1 m
Used in radio and television communication
systems
Microwaves



Wavelengths from about 0.3 m to 10-4 m
Well suited for radar systems
Microwave ovens are an application
Notes on the EM Spectrum, 2

Infrared waves





Wavelengths of about 10-3 m to 7 x 10-7 m
Incorrectly called “heat waves”
Produced by hot objects and molecules
Readily absorbed by most materials
Visible light


Part of the spectrum detected by the human
eye
Most sensitive at about 5.5 x 10-7 m (yellowgreen)
More About Visible Light


Different
wavelengths
correspond to
different colors
The range is from
red (l ~7 x 10-7 m)
to violet (l ~4 x 10-7
m)
Visible Light – Specific
Wavelengths and Colors
Notes on the EM Spectrum, 3

Ultraviolet light




Covers about 4 x 10-7 m to 6 x10-10 m
Sun is an important source of uv light
Most uv light from the sun is absorbed in the
stratosphere by ozone
X-rays



Wavelengths of about 10-8 m to 10-12 m
Most common source is acceleration of highenergy electrons striking a metal target
Used as a diagnostic tool in medicine
Notes on the
EM Spectrum, final

Gamma rays




Wavelengths of about 10-10m to 10-14 m
Emitted by radioactive nuclei
Highly penetrating and cause serious
damage when absorbed by living tissue
Looking at objects in different portions
of the spectrum can produce different
information
Wavelengths and Information


These are images of
the Crab Nebula
They are (clockwise
from upper left)
taken with




x-rays
visible light
radio waves
infrared waves
Polarization of Light Waves



The electric and magnetic vectors associated
with an electromagnetic wave are
perpendicular to each other and to the
direction of wave propagation
Polarization is a property that specifies the
directions of the electric and magnetic fields
associated with an em wave
The direction of polarization is defined to be
the direction in which the electric field is
vibrating
Unpolarized Light, Example




All directions of
vibration from a wave
source are possible
The resultant em wave
is a superposition of
waves vibrating in many
different directions
This is an unpolarized
wave
The arrows show a few
possible directions of
the waves in the beam
Polarization of Light, cont


A wave is said to be linearly
polarized if the resultant
electric field vibrates in the
same direction at all times at
a particular point
The plane formed by the
electric field and the direction
of propagation is called the
plane of polarization of the
wave
Methods of Polarization


It is possible to obtain a linearly
polarized beam from an unpolarized
beam by removing all waves from the
beam expect those whose electric field
vectors oscillate in a single plane
The most common processes for
accomplishing polarization of the beam
is called selective absorption
Polarization by
Selective Absorption

Uses a material that transmits waves whose electric
field vectors in the plane parallel to a certain direction
and absorbs waves whose electric field vectors are
perpendicular to that direction
Selective Absorption, cont

E. H. Land discovered a material that
polarizes light through selective
absorption


He called the material Polaroid
The molecules readily absorb light whose
electric field vector is parallel to their
lengths and allow light through whose
electric field vector is perpendicular to their
lengths
Selective Absorption, final


It is common to refer to the direction
perpendicular to the molecular chains
as the transmission axis
In an ideal polarizer,


All light with the electric field parallel to the
transmission axis is transmitted
All light with the electric field perpendicular
to the transmission axis is absorbed
Intensity of a Polarized Beam

The intensity of the polarized beam
transmitted through the second
polarizing sheet (the analyzer) varies as

I = Io cos2 θ


Io is the intensity of the polarized wave incident
on the analyzer
This is known as Malus’ Law and applies to any
two polarizing materials whose transmission
axes are at an angle of θ to each other
Intensity of a
Polarized Beam, cont

The intensity of the transmitted beam is
a maximum when the transmission axes
are parallel


q = 0 or 180o
The intensity is zero when the
transmission axes are perpendicular to
each other

This would cause complete absorption
Intensity of
Polarized Light, Examples



On the left, the transmission axes are aligned and
maximum intensity occurs
In the middle, the axes are at 45o to each other and
less intensity occurs
On the right, the transmission axes are perpendicular
and the light intensity is a minimum
Properties of Laser Light

The light is coherent



The light is monochromatic


The rays maintain a fixed phase relationship with
one another
There is no destructive interference
It has a very small range of wavelengths
The light has a small angle of divergence

The beam spreads out very little, even over long
distances
Stimulated Emission



Stimulated emission is required for laser
action to occur
When an atom is in an excited state, an
incident photon can stimulate the electron to
fall to the ground state and emit a photon
The first photon is not absorbed, so now
there are two photons with the same energy
traveling in the same direction
Stimulated Emission, Example
Stimulated Emission, Final



The two photons (incident and emitted)
are in phase
They can both stimulate other atoms to
emit photons in a chain of similar
processes
The many photons produced are the
source of the coherent light in the laser
Necessary Conditions
for Stimulated Emission


For the stimulated emission to occur,
there must be a buildup of photons in
the system
The system must be in a state of
population inversion


More atoms must be in excited states than
in the ground state
This insures there is more emission of
photons by excited atoms than absorption
by ground state atoms
More Conditions

The excited state of the system must be
a metastable state



Its lifetime must be long compared to the
usually short lifetimes of excited states
The energy of the metastable state is
indicated by E*
In this case, the stimulated emission is
likely to occur before the spontaneous
emission
Final Condition

The emitted photons must be confined



They must stay in the system long enough
to stimulate further emissions
In a laser, this is achieved by using mirrors
at the ends of the system
One end is generally reflecting and the
other end is slightly transparent to allow
the beam to escape
Laser Schematic

The tube contains atoms



The active medium
An external energy source is needed to “pump” the
atoms to excited states
The mirrors confine the photons to the tube

Mirror 2 is slightly transparent
Energy Levels, He-Ne Laser




This is the energy level
diagram for the neon
The neon atoms are
excited to state E3*
Stimulated emission
occurs when the neon
atoms make the
transition to the E2 state
The result is the
production of coherent
light at 632.8 nm
Laser Applications



Laser trapping
Optical tweezers
Laser cooling

Allows the formation of Bose-Einstein
condensates