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Lecture 19: Interference
When two electromagnetic waves meet, the electric fields add
up with sign. Namely they can reinforce (constructive
interference), they can cancel each other (destructive
interference), or anything in between.
Suppose the electric fields have the same strength:
Constructive, maxima coincide:
Destructive, maximum coincides with minimum:
Random: Sometimes constructive, sometime destructive, in
average the intensity adds up, namely it is double. This is what
happens in most everyday situations. The reason is that light
is made out of a series of wave packets.
Wave packets interfering:
Length of packet=coherence
length
Examples:
White light: ~ 1mm
Single spectral line: ~ 1m
Lasers: 10 cm to 10 Km.
The only practical way to observe interference is to use light from
the same source by splitting and rejoining a beam.
Interferometer:
LIGO Interferometer to detect gravitational waves.
Washington state site:
Main rule of interference:
Here DL is the difference in lengths of the two paths the light takes.
In the interferometer it is DL=2 (L2-L1).
We also need DL > l where l is the coherence length.
A very important point is that the wave-length is determined by the
colors. Different colors have different wave-length.
Example: lyellow= 570nm, lgreen=510nm, lblue= 475nm
This means that the same setup can give a maximum for one color
and a minimum for another, colors separate. In everyday life this
is seen for example in soap bubbles, oil films on water, peacocks.
Thin film interference:
We could now use the main rule of interference:
But there are two important subtleties.
1) The wave length depends on the medium. Frequency is the
same.
2) There can be a phase shift when light from medium 1 is reflected
by an interface with medium 2 (same as extra half wave-length).
As a result we find the rule for thin film interference:
Newton’s rings. Consider an analogous of the Newton rings by
putting to pieces of glass at an angle using a thin wire to separate
them at the end. Newton rings are done by putting a lens over a
flat glass.
Maxima
Clicker question
In the previous set up L=15cm, h=0.24mm,
l=600nm.
How many fringes (bright bands) do you see
A.
B.
C.
D.
400
600
800
1000
Two slit experiment
Bright bands (fringes) for
those angles such that
Lecture 20: Fresnel equations
Reflection and refraction. Assume E parallel to interface.
Continuity of parallel Electric field
Continuity of Magnetic field
Standard relation for electromagnetic waves
We derive the following equations for the electric field:
But:
So:
Remember:
Then
,
Snell’s law!!
But we have one more equation
Given incident we can find transmitted and reflected.
Consider normal incidence
For general angle
Again:
In any case, how much is reflected and how much refracted is
completely determined by these equations!!.
Lecture 21: Diffraction
Gratings
We can reinforce the effects of two slits interference by including
many slits. This is called a grating and has much sharper maxima.
Maxima whenever
Application: To separate the color of an incident light. The same
effect can be obtained with a prism, however a prism is a thick
(triangular section) slab of glass and absorbs too much light for
certain applications (for example star light).
The spectrum of distant stars can be obtained by using a grating. It
turns out that different elements have a special signature in the
spectrum (certain wave-lengths) and therefore the composition
of far away stars can be determined. In fact Helium was
discovered first in the Sun!!.
Diffraction
A particular case of interference is when light goes thought a hole
and it has its own name: diffraction.
In this case it is useful to look at the minima. Whenever a ray in the
top interferes destructively with a ray in the middle, the same will
be true for every ray in the upper half, namely there will be
another ray which cancels it.
First minimum
Other minima are at qn=n l/a, where n is an integer (order).
If we put a screen a distance D away the width of the bright band in
the middle is:
The width of the other
bright fringes is half the
width of the central one.
Important effect. Parallel rays going through a hole are not parallel
any more afterwards. They have a range of angles from zero to
For a round aperture hole the formula is
In the case of a convergent lens, parallel rays produce an image of
size
In fact the image is a solid dot surrounded by diffraction rings.
Observing those rings is the indication that this particular
aberration is the effect of diffraction. It limits the resolution of
optical instruments including our eye!. For ground based
telescopes, the resolution is actually given by the atmospheric
fluctuations since they are large enough.
For the eye, assuming a 4mm pupil:
in radians.
Did Nature gave us the maximum possible resolution for our eyes size?.
Clicker question
Assume that the resolution of the eye is indeed the
maximum allowed by diffraction. If the two
headlights of a car are 1m apart, at what
maximum distance would you be able to resolve
them?
A. 1 Km
B. 10 Km
C. 100 Km
D. 1000 Km
Lecture 22: Light-Matter interaction
Photoelectric effect
Light is emitted and absorbed by matter. At the end of the 1900
this is was seen as one of the few phenomena left to
understand in physics. It turned out to cause a revolution in
physics that we still live today.
Photoelectric effect:
Electrons are free to move inside a metal but normally do not
leave it since the metal would be positively charged an attract
them back. Light has a powerful localized electric field that
can kick the electrons far enough so they do not come back to
the metal. Once electrons are away one can direct them with
electric fields. Certain night vision systems for example work
that way.
In physics we are concern with two main quantities, how many
electrons are kicked out and with which energy. The number of
electrons, not surprisingly, is proportional to the intensity of
light. But their energy is proportional to the frequency!
The surprising explanation was proposed by Einstein. He proposed
that light is made out of quanta, or particles, called photons
whose energy is given by (h is called Planck’s constant)
Now the explanation is very simple. A photon is absorbed by an
electron which acquires an extra energy Eph. Part of the energy is
used to leave the metal (called W, work function), the rest
becomes kinetic energy.
We defined
Now, it follows that
This provides a simple explanation of the experimental results. To
be truthful to Einstein genius, however, we should point out that
the formula was derived by Einstein nine years before the first
experiment that actually verified it!. (It was known that the
energy of the electrons increased with frequency but not this
simple formula).
Photons
The photoelectric effect shows that light interacts with matter as if
made of tiny quanta called photons. They have energy and
momentum given by
With h c = 1.2 x 10-6 eV m (c is the speed of light)
An electron Volt, eV is the energy gained by an electron when going
through a potential difference of 1 V.
1eV = 1.6 x 10-19 J
Clicker question
Light of wave-length l=100nm shines on Copper,
W ~ 5eV (more precise value 4.7 eV).
What’s the energy of the ejected photons?
A.
0.7 eV
B.
7 eV
C. 70 eV
D. 700 eV
hc = 1.2 x 10-6 eV m
Eph = h f = hc / l
Ee = Eph -W
Atomic theory.
The next step in understanding the interaction of light with matter
is the emission of light by an atom. The simplest atom is
Hydrogen made of a proton and an electron. It turns out that a
hydrogen atom emits (and absorbs) only particular frequencies, it
has a sort of spectral fingerprint. The same is true for all
elements.
Explanation (Bohr). The electron orbits around the proton in some
specified orbits. The atoms absorbs and emits photons when the
electron jumps from one orbit to another.
This explains the resulting spectrum but goes against Newtonian
mechanics. There are two main problems.
1) Why does not the electron find its minimum of energy by
attaching to the proton, namely why there is a lowest energy
orbit?.
2) Why only certain orbits are possible?
It means that new physical principles operate at distances of the
atomic size. A new length scale brings in new physics!
1) Is explained by Heisenberg’s uncertainty principle.
2) Is explained by De Broglie’s particle-wave duality.
In fact it is necessary to develop a new formalism called quantum
mechanics. We are not going to go that far here.
Uncertainty principle.
In classical mechanics the state of a particle is determined by its
position and its momentum (or velocity).
In quantum mechanics the position and momentum cannot be
determined simultaneously. If the particle is localized, the
momentum is undetermined and vice-versa. The uncertainty
obeys:
Consider the hydrogen atom. The energy is
The potential energy is minimized when the electron is on top of
the proton (r=0). However localizing the electron would make the
momentum completely uncertain and therefore the energy
would be huge (in Newtonian mechanics, instead, we can have
r=0, p=0) . We estimate
We get
where we defined
or
The energy is a function of the orbital radius and becomes a
minimum for r = 2 r0. This gives a correct lowest energy (n=1)
Lecture 22: De Broglie waves.
Light behaves as a wave but also as particles.
Can particles such as the electron behave as waves? Yes, that was
the proposal of De Broglie. The frequency and wave-length are
given by:
We also have the usual relation between momentum and energy
where we included the comparison with the case of the photon.
This received truly spectacular experimental confirmation by an
analogous of the two slit experiment:
Electron
source
The interference pattern is
present even if we shoot the
electrons one by one!!
Quantum mechanically the
electron explores all possible
paths.
Clicker question
An electron is moving at 30 m/s. What is the
associated wavelength?.
A.
B.
C.
D.
2.4 x 10-5 m
2.4 x 10-7 m
2.4 x 10-9 m
2.4 x 10-11 m
hc = 1.2 x 10-6 eV m
Pe = me v = h / l
me c2 = 0.5 MeV
How can this help us understand the Hydrogen atom?.
The orbits are standing waves where an integer number of wavelengths fit.
Let’s see. Use Newtonian relation to get the velocity
and then the momentum
to find the possible orbits
Spectacular agreement!!
Energy:
Quantum mechanics is the basis for our understanding of the
microscopic world. It replaces Newtonian mechanics at the
atomic level. It is fundamental to understand properties of
materials (conductors, insulators, semiconductors,
superconductors,…). At smaller length scales, nuclear and
particle physics are based on quantum mechanics. No deviations
of quantum mechanics have been found down to distances
d~10-20m (LHC accelerator).
Future? Quantum computers is an active topic of research and can
revolutionize computations. Quantum cryptography. New
material, new states of matter (Bose-Einstein condensate), etc.
Although a hundred years old, quantum mechanics is present
everywhere in the frontier of research and technology.
Example: Lasers (and masers). It is based on the effect that an
excited electron is more likely to emit a photon in a photonic
state that is already occupied. It is called stimulated emission and
can be used to amplify an already present wave.
It can also be used for microwaves. It is the basis of atomic clocks
which are used to synchronize GPS satellites.
Example: Compton effect. Energetic photons can bounce form
electrons losing energy in the process.
The precise computation requires using special relativity to describe
the electron but the result is very simple:
Lecture 24: Nuclear Physics
Atomic physics involve distances ~10-10 m and energies 10 eV.
Nuclear physics operate at distances 10-15 m and, by the
uncertainty principle we expect it involves energies 105 larger,
namely E~1MeV.
The nucleus is composed of proton and neutrons. Protons are
positively charged and neutrons are neutral. The number of
protons is called Z, the atomic number. The total number of
protons and neutrons is called A, the mass number. The
number of neutrons is therefore A-Z.
The neutral atom has Z electrons which determine the chemical
properties of the element. Therefore Z determines which type
of element the atom is (Hydrogen, Carbon, Oxygen, etc.).
The number of neutrons is not apparent in chemical properties
and can vary for the same chemical element. The main effect
is that the mass of the atom is different.
Nuclei with the same Z but different A are called isotopes. It is
difficult to separate isotopes because it cannot be done
chemically. It can be done usually using the different
densities, diffusion constants etc.
Notation:
A
Z
Z
Isotope of Carbon. 6 protons, 8 neutrons. Most abundant is
Carbon 12, it has 6 protons and 6 neutrons.
The mass of the proton and neutron are close to each other and
to the value m~1.6x10-25 Kg
The mass of the nucleus is approximately equal to the sum of the
masses of protons and neutrons that it contains. However precise
measurements determine a mass defect:
This is actually related to the binding energy by the famous Einstein
formula:
This is always true but the energies involved in nuclear physics are
large enough (and the mass can be measured with enough
precision) that Einstein formula can be verified. We need
however more precise data for the masses of protons and
neutrons:
The binding energy comes from the nuclear force. Protons and
neutrons attract each other with a force larger that the electric
repulsion (between protons).
Nuclei can either spontaneously or by external influence, decay,
namely be converted into other nuclei.
1) Alpha decay. A nucleus of Helium is a particularly stable nucleus
called an alpha particle. Some nuclei decay by emission of alpha
particles:
Example Radium (Ra)
2) Beta decay. Free neutrons are actually unstable and decay into
protons emitting an electron and a neutrino. Electrons are called
beta particles in this context.
Inside a nucleus this is usually not energetically favorable so it does
not happen. However some nuclei are unstable
Notice that one neutron was converted into one proton.
This is the effect of the weak force. In this particular case it is used
for Carbon dating.
Other cases:
Electronic capture:
An orbital electron is captured by the nucleus. A proton becomes
neutron. X-rays are emitted when other electrons fill the hole left
by the captured electron.
Finally we can have inverse beta decay where a proton becomes a
neutron and emits a positron:
3) Gamma decay. This is a standard electromagnetic transition
where, for example, a nucleon changes orbit emitting a photon,
the same as for atoms. The photon in this context is called a
gamma ray and has E~ 1 MeV.
These photons are actually the most dangerous products of nuclear
decays and are responsible for the “danger radioactivity signs”.
Alpha particles and electrons are easily stopped by matter.
Gamma rays are not, for example lead bricks are used to
(partially) stop them.
The dose of absorbed radiation is measured in Gray (Gy).
1 Gy = 1 J/ Kg = (energy absorbed in Joules)/(mass in kilograms)
For any of those decays, we start with a number N of nuclei. After a
while a fraction of those decayed. The time it takes for half of
them to decay is called the half-life T1/2. The number of nuclei in
the original states decreases as
For example the half-life of
is 5730 years.
The ratio of
to
is the same as in the environment as
long as a organism is alive. After death, the amount of
begins to decrease and can be used to date the (archeological)
remains.
Another way to characterize how fast a sample is decaying, and
therefore know for example the intensity of the radiation that it
can produce is by counting the number of decays in a second.
This is known as the activity of the radioactive source and
depends on the half-life of the nucleus and also on how many
nucleus are present. It is measured in Curies or Becquerels:
The other two important processes we need to consider are fission
and fusion. Fission is the split of the nucleus into two nuclei of
approximate half the size. This can be spontaneous but most
commonly occurs by neutron bombardment. Example:
The most common form of Uranium is actually
which
does not undergo fission. The only way to get a sustainable rate of
fission (or explosive) is by increasing the concentration
a process called uranium enrichment. This is a source of energy
Both, in nuclear reactor and nuclear explosives. For example 1g of
Uranium completely undergoing fission releases 9x1010J to be
compared with 1g of TNT which produces 4200 J.
Finally we have fusion where two nuclei combine to form a larger
one. This is actually how the elements in the Universe were
produced (in stars).
Example:
These are different isotopes of hydrogen actually. They are called
deuterium and tritium. Fusion also releases large quantities of
energy and is the energy source of the Sun and the basis for
fusion weapons. It very difficult to control in a way that it
produces more energy than the one used to produce it. If that is
achieved it will be the source of energy of the future. There is a
large international collaboration attempting this (ITER). See also
National Ignition Facility, Livermore, CA (DOE).