Chapter 23: Electromagnetic waves
Arecibo 305m radio telescope
Contents:
Speed of EM waves
The EM spectrum
Transmitted energy
Nature of light
Reflection and Refraction
Total internal reflection
Polarization
Huygens’ principle
Scattering of light
Convention:
In this lecture EM = electromagnetic.
solar spectrum
What will we learn in this chapter?
Brief introduction
Recall Faraday’s law:
!
!
!
!
A changing magnetic field B (flux) induced an emf in a circuit:
!
!
! ∆ΦB !
!
E = !!
∆t !
An induced emf produces an electric field E.
This suggests that time-dependent electric and magnetic fields are
intrinsically related, thus we can use the term electromagnetic field.
We can assume that energy travels in waves, like ripples in a pond.
This is summarized in Maxwell’s equations:
!E
! = ρ/#0
! ×E
! = − ∂B
∇
∇
∂t
! ×B
! = µ0 "0 ∂E + µ0!j
!B
! =0
∇
∇
∂t
Speed of an electromagnetic wave
Setup:
Assume we have an E-field in the z-direction and a B-field in the
y-direction producing an electromagnetic wave traveling in the
x-direction.
Rudimentary EM wave:
The yellow plane corresponds
to the wave front traveling with
a speed c.
In front of the wave front, there
is no field.
Behind the wave front, E and
B fields are perpendicular
to each other.
Speed of an electromagnetic wave contd.
Conditions for the rudimentary wave to be an EM wave:
At any point in time, the magnitudes of E and B have to satisfy:
E = cB
The wave front moves with a speed c given by
1
c= √
!0 µ0
where
!0 = 8.8542 × 10−12 C2 /Nm2
µ0 = 4π × 10−7 N/A2
Note that c = 299792458 m/s is the speed of light in vacuum!
Note:
By combining two of Maxwell’s equations, we obtain the wave
equation (harmonic motion) with the aforementioned speed.
Characteristics of EM waves in vacuum
Conditions a EM wave has to fulfill:
The wave is transverse, i.e., both E and B are perpendicular to the
direction of propagation and to each other.
There is a definite ratio between the magnitudes: E = cB
The wave travels in vacuum with a constant speed c.
The wave does not need a medium to propagate. The fields
oscillate.
Modified right-hand rule for EM waves:
Point the right thumb in the travel
direction of the wave.
Index fingers points in E direction.
Middle finger points in B direction.
The electromagnetic spectrum
long wavelength
short wavelength
EM waves cover a broad spectrum of wavelengths λ, but all
propagate with the same speed. Hence c = λf.
The visible light only covers a narrow range of the spectrum.
Although we cannot see outside the visible range, there are machines
and animals that can.
what we see
The Milky Way in different wavelengths
Nature and “invisible” spectrum ranges
Pit vipers:
Can see in the infra-red spectrum.
Detect heat generated by prey and hunt
“blindly.”
pit viper
Bees:
Can see in the ultraviolet.
Some plants “advertise” in this
wavelengths.
UV
Shell parakeets:
Females determine
the nicest mate in
the UV range.
visible
UV
visible
Harmonic waves
Harmonic EM waves are analogous to transverse mechanical waves in
stretched strings.
In an EM wave, the E and B field are sinusoidal in time.
This is similar to the rudimentary wave
in that the fields are uniform in the
propagation front. We call such a
wave a plane wave.
EM waves can be described by
wave functions:
!
"
t
x
y(x, t) = A sin 2π
−
= A sin(ωt − kx)
T
λ
where y is the transverse displacement at time t of a point with
coordinate x on a string. A is the amplitude, ω = 2πf and k = 2π/λ
is the wave number.
Harmonic waves contd.
We can now represent the EM fields as follows:
!
"
t
x
E = Emax sin 2π
−
= Emax sin(ωt − kx)
T
λ
!
"
t
x
B = Bmax sin 2π
−
= Bmax sin(ωt − kx)
T
λ
The amplitudes must be related via
Emax = cBmax
Note:
A wave whose E-field always lies
along the same line is said to be
linearly polarized.
In the example the wave is
linearly polarized in the y direction.
Energy in electromagnetic waves
Energy can be associated with electromagnetic waves. Examples:
Sun (heats up planet)
Microwave oven (resonance with water, heats food)
X-rays
Energy density of electric and magnetic fields: The energy density u
(energy per volume) in a vacuum with electric and magnetic fields
present is
1
1 2
u = !0 E 2 +
B
2
2µ0
Note:
√
Use B = c−1 E = !0 µ0 E and replace the B field by an E field. It
follows that u = !0 E 2. The energy density of both fields is the
same.
Since EM waves travel, it follows that they transfer energy.
Transferred energy
Traveling EM waves transport energy in space.
Intensity of a wave:
Transferred power per unit area.
Study a wave in x direction which moves
by ∆x = c∆t .
Consider an area A on the wave front.
The energy in that region is the volume
∆V = A∆x times the energy density:
∆U (∆t) = u∆V
∆U = u∆V = (!0 E 2 )(Ac∆t)
The energy flow per area A and time
∆t thus is
1 ∆U
S=
= !0 cE 2
A ∆t
This can be rewritten…
Transferred energy contd.
We can also derive the following alternative forms:
1
EB = cu
µ0
The units of S are power per unit area [J/(sm2)].
Note:
So far we have only treated instantaneous values of the fields.
For a harmonic wave the average value of E2 is 1/2 the amplitude.
!
It follows:
1 !0 2
1
1
2
Sav =
Emax = !0 cEmax
Emax Bmax
=
2 µ0
2
2µ0
Similarly to previous calculations we obtain for a sinusoidal wave
uav
Sav = √
= uav c
!0 µ0
which is also known as the Intensity I = Sav.
S = !0 cE 2 =
Radiation pressure
The fact that EM waves transport energy follows from the fact that
energy is required to establish the E and B fields.
It can also be shown that EM waves carry momentum p, with a
corresponding momentum density
p
!0 E 2
EB
S
=
=
=
V
c
µ0 c2
c2
For harmonic waves we can look at the average momentum:
2
pav
!0 Emax
Emax Bmax
Sav
=
=
=
V
c
µ0 c2
c2
and the momentum flow ∆p in a time interval ∆t trough A is
1 ∆p
1
Sav
I
= !0 E 2 =
=
A ∆t
2
c
c
Note:
The EM momentum is a property of the field and not associated
with any moving particle masses.
Radiation pressure contd.
The momentum transfer is responsible for radiation pressure:
A propagating wave excerpts a force on a surface.
The average pressure is I/c.
If the wave is totally reflected then the pressure is
2I/c.
Examples:
The radiation pressure of sunlight on an absorbing surface is
approximately 4.7x10-6 Pa (this can be measured!).
Theoretically, one could use the pressure of
the sun’s EM waves to power spacecrafts–so
called sun sails. Some projects are currently
being explored.
Tails of comets always face away from the
sun due to the sun’s radiation pressure.
Nature of light
Is it a particle? Is it a wave? Is it a ray? It’s all of them!
Rays:
Reflection, refraction,
geometrical optics, …
Wave:
Diffraction
Particle:
The emission/absorption of light by atoms suggests that light also
has a particle character (quanta of energy).
We call these photons.
EM radiation (light) sources
Electric charges in accelerated motion:
X-ray machines
Thermal radiation due to molecular motion:
Flames & coals in a camp fire
Hot metallic objects appear red hot
Light bulbs (the old kind)
Electrical discharge in ionized gases:
Sodium lamps (crucial for astronomers)
Neon signs
Lasers:
Monochromatic light sources
Wave fronts
Definition (wave front): ! Locus of all adjacent points at which the
! ! ! ! ! ! ! phase of vibration of the wave is the same.
Examples: ripples in a pond, sound waves.
The wave crests are separated from each
other by one wavelength.
For EM waves the displaced quantity is
the electric or magnetic field:
point source
far away from the source
Wave fronts contd.
It is often convenient to represent the
light wave as light rays instead of wave fronts.
In fact, early descriptions of light assumed
particles traveling along straight paths.
A ray is an imaginary line along the
direction of travel of the wave.
Rays can change direction when different mediums come together.
This will be discussed next within the ray model of light propagation.
Reflection and refraction
Common scenario:
Different images of the hat can be seen.
Reflection and refraction contd.
Analysis:
Light strikes the smooth glassy
surface and changes medium.
Part of the wave is refracted
(transmitted).
Part of the wave is reflected.
Simplified problem:
It is enough to study the
behavior of one representative
ray of light.
One can define angles between
incident/reflected/refracted rays
with respect to the Normal.
Specular vs diffuse
If the interface is rough, both transmitted and reflected light are
scattered in various directions and there is no single angle of
transmission or reflection.
Specular reflection:
On a smooth surface
There is a definite angle
Diffuse reflection:
Rough surface
Scattered reflection
No definite angle
Index of refraction
Material-dependent quantity which plays a central role in geometric
optics.
Index of refraction: The index of refraction of an optical material (n) is
the ratio of the speed of light in vacuum (c) to the speed of light in the
material (v):
c
n=
v
For vacuum n = 1 by definition.
Note:
In general, light travels slower in a material hence n > 1.
As we will see later, n can be determined geometrically as well.
Principles of geometric optics
The incident, reflected, and refracted rays, and the normal to the
surface, all lie in the same plane.
The angle of reflection θr is
is equal to the incidence angle
θa for all wavelengths and for
any pair of substances:
θa = θr
For monochromatic light
passing from medium a to
medium b the angle of refraction
is given by
sin θa
nb
=
(Snell’s law)
sin θb
na
Snell’s law: some considerations
A ray passes from material a to material b…
nb > na
The refracted beam bends
toward the normal.
nb < na
The refracted beam bends
away from the normal
If the angle of incidence is
zero, the rays is not bent.
Snell’s law contd.
The path of the refracted ray is
always reversible.
The ray follows the same path
when going from b to a as when
going from a to b.
The path of the reflected ray is
also always reversible.
The intensities of the reflected and
refracted rays depend on the angle
of incidence, the indices of refraction
and the polarization of the incoming
ray. Reflection is smallest when the
angle of incidence is zero.
Index of refraction for yellow sodium light
λ = 589nm
Dispersion
The index of refraction depends
on the wavelength of the incident
light.
This is called dispersion.
This is how rainbows form and why
prisms produce mini rainbows.
Wavelengths at interfaces
When light passes from one material to another, the frequency f
does not change.
Reason:
The boundary cannot create/destroy waves.
The number arriving per unit time must be the same as the
number leaving per unit time.
In any material v = λf . Because f is fixed and v = c/n it follows that
in a medium
λ0
λ=
n
where λ0 is the vacuum wavelength.
Total internal reflection
Rays are emitted from a point P
in a material a and strike the
surface of a material b with nb < na.
Because na/nb > 1 there must be a
critical angle θa for which θb = 90◦.
The critical angle for which light
travels along the interface is given
by
nb
sin θcrit =
na
For larger angles than θcrit the
sine is not properly defined, the
light is trapped in the lower material and reflected back.
Total internal reflection: If nb < na a ray is reflected back into the
material if the angle of incidence is greater than θcrit = arcsin(nb /na ).
Total internal reflection contd.
For a glass-air surface θcrit = arcsin(1/1.52) = 41.1◦.
The fact that this angle is smaller than 45º makes
it possible to use triangular prisms as a totally
reflecting surface. This has advantages over metallic
mirrors since the latter oxidize and do not reflect
100%.
This technology is commonly used in optical
instruments:
Porro prism
binoculars
Total internal reflection contd.
Optical fibers:
When light enters a transparent rod which is not curved too
strongly it is trapped (optical fibers).
Technological applications:
Data transfer
Endoscopes
International radio transmission:
Thanks to total reflection on
the atmosphere we can
tune into Deutsche Welle
in the USA.
Texas
Or just listen online…
DW
Polarization (mechanical analogy)
Definition:
When a transverse wave has only displacement in the !-axis we
say the wave is !-polrized.
Examples:
linearly-polarized in the y-direction
linearly-polarized in the z-direction
The wave will oscillate in the x-y or x-z plane, respectively.
Polarizing filters:
Permit motion only in a given
direction.
Polarizing filters for EM waves
Construction:
Depends on the wavelength range to be polarized. Examples…
Microwaves:
A grid of closely-spaced, parallel conducting wires.
! ⊥ wires the field passes.
unpolarized light
If E
Optical:
Polaroid filters ® use
materials that exhibit dichroism
(anisotropic absorption of horizontal
polarization
polarizing components).
absorbed
80% of the intensity is
transfered parallel to the
polarizing axis, less than 1%
in other directions.
filter
linearly-polarized light
Polarized EM waves
Ordinary light sources:
Light is emitted by many “out of sync” molecules. Thus light is
unpolarized.
Radio transmitter antennas:
Usually linearly polarized (vertically in
the US, horizontally in the UK).
Material science:
Can be used to
detect stress in
materials.
Later…
Polarizing light
The intensity of the transmitted light is the
same for all filter orientations. Why?
The incident E-field can be described as parallel
and perpendicular to the polarization axis.
When we measure the intensity (power/area) only 50% get
trough since the incident light is randomly polarized and thus
the components of the field are on average equal.
Place a second polarizer before photocell
angle between polarizer
and analyzer
maximal intensity for zero angle
linear polarized
light can be divided
into components
Once the light is linearly polarized it can be “cleanly” divided into
components. Only the parallel component E|| = E cos φ is
transmitted.
The intensity of EM waves is proportional to the square of the
amplitude. It follows…
Polarizing filters contd.
Light transmitted by polarizing filters (Malus law): When linear-polarized
light strikes a polarizing filter with its axis at an angle φ to the direction
of the polarization, the intensity of the transmitted light is
I = Imax cos2 φ
with Imax the maximum intensity at φ = 0 .
The effects of polarizing
filters are nicely illustrated by
the pair of sunglasses.
No transmission for 90º.
The reason sunglasses have
polarizing filters is to
eliminate glare from asphalt
surfaces.
φ = 90◦
φ=0
Polarization by reflection
Unpolarized light can be partially polarized by reflection on a surface
between two optical materials if it strikes the surface at an angle θp.
Parallel to surface is refracted.
Refracted light is slightly polarized in the plane of incidence.
Polarization by reflection: Brewster’s law
When lights strikes a surface at a particular angle θp, the field
perpendicular to the plane of incidence is reflected and thus
polarized in that direction (parallel to the surface).
component perpendicular to page
When light at an angle θp hits the surface,
none of the E-field parallel to the plane
of incidence is reflected; it is transmitted
100% in the refracted beam.
While the reflected light is polarized,
the refracted component is partially
polarized.
When the light hits the surface at θp ,
refracted and reflected beams are
perpendicular to each other.
Brewster’s law:
nb
tan θp =
na