Download The Demonstration

These are all the lectures that we have. I slapped them all together kinda hastily, so just
email me if there are any problemos.
2-1
2-27
3-1
3-3
3-6
3-10
3-13
3-24
4-3
4-5
4-7
4-10
4-12
4-14
4-17
4-19
4-21
4-24
4-26
4-28
5-3
2/1/06 SCA-29 Lecture
First lecture of the course. He spends some time going over the mechanics of the court.
Light Demonstration:
He shows us the visible spectrum of light (the rainbow) projected on a white wall. There
is only three colors visible to us: red, green and blue. As far as vision is concerned, we
can only observe these three colors. He mentions the invisible spectrum on the right and
left of the visible spectrum. The visible spectrum is the tiniest representation of all the
colors that are possible when we consider infrared and ultraviolet spectra.
Periodic Demonstration:
He has six “duck-like” contraptions under a heat lamp that move back and forth.
Light Demonstration:
Light gives us a way of making extraordinarily sensitive measurements. He shines a laser
beam onto a mirror. He is detecting that laser beam in a very sensitive way. It can detect
the tiniest movements of light. So when he presses on the wall that the mirror is attached
to we hear an audible “wooshing” sound. This phenomena is based on the wave nature of
light.
Electricity & Magnetism:
He has one of those bulbs with the light emanating from a type of tesla coil: A high
frequency electrical discharge in a gas. When he touches the globe and the light’s
“fingers” touch his own.
Tesla Coil:
He has a mini tesla coil that generates a bluish/white light (and makes quite a noise when
he turns it on). He holds a light tube near the coil and we see a current leap from the coil
into the bulb/tube and it lights up.
These are all glimpses of what we will study in the course of the semester.
But for now he wants to impress on us that light travels in straight lines. He holds up a
laser beam to make his point. He also turns on a smoke generator, which makes it
possible for us to see the beam of light from its origin to its destination.
He has a stereophonic recording, which basically means that sounds that are louder in one
microphone than the other. He has set up a red laser beam, and a pair of mirrors that are
held at angles such that the signal that comes from one of the two loudspeakers displaces
the beam vertically and the other track gives a horizontal displacement to the beam.
When music is played these two signals are going on simultaneously. As he plays this
futuristic piece the laser scribbles all over the screen.
SCA29 Lecture – 2/27/06
Demonstration #1: Polarization
He has an aquarium tank.The tank has a beam of light projected into it from a light
projector, and there are mirrors.
He holds up a polarization filter to show scattering.
Then he adds some milk to one of the tanks.
The dramatic darkening effect of the polarization filter is due to the scattering effect of
the tiny particles of milk. You can actually see the little globules of milk through the
polarization filter.
Then he adds a lot more milk, which will make the scattering more intense. He notes that
the more milk you add, the less fractual polarization you get. When you add a fair amount
of milk, you begin to get multiple scatterings. That is to say one light ray scatters from
one particle of milk to another to another to another, zig-zagging it’s way through the
medium. In this case you don’t get the same amount of polarization.
The Polarization is most dramatic when you have only single scattering processes taking
place.
If he polarizes the light going in, he is sending in horizontally polarized light. That light
gets scattered in a single horizontal plane. Scattering is less intense.
Demonstration #2: Polarization
He has a laser that turns out a green polarized light beam. He sends this beam into a long
cylindrical tank that has a sugar solution in it. The sugar molecule has a structure that
curiously tends to rotate the direction of the polarization of light as it passes through the
sugar solution. There is scattering that’s taking place, but the polarization of the beam
going through the tank is being rotated as the beam moves along the tank. The scattering
is always the strongest when the direction of polarization is perpendicular to your
line of sight. It is a strange thing to shine a beam of light through a solution that’s
essentially uniform and seeing the scattering taking place intermittently.
More strange things that one can do with polarized light.
Isotropic (materials like plastics that have the properties relative to the direction of light
propagation) and Anisotropic (crystalline materials that have different properties relative
to the direction of light propagation).
Some isotropic materials (he holds up a piece of plastic in the shape of a V and squeezes
it and changes the optical properties of this piece of plastic by squeezing). What is
happening here is an isotropic substance is being made anisotropic by squeezing. What’s
happening is that this piece of isotropic plastic is being rendered anisotropic by
squeezing which rotates the direction of the polarization of light going through it.
That rotation of the polarization of the light makes it so that the light comes through over
the projector. Rotating the direction of polarization is the purpose of this
demonstration. He uses a number of plastic discs to demonstrate this property. He puts
them on the projector (which polarizes light, and he also holds up another filter that
changes the polarization). He then holds up a block of plastic which is being squeezed in
a vice, and when he holds the other filter up, you can see how the stress changes the
rotation of the polarization of light.
Cellophane example: it was the first flexible transparent wrapping material that was
developed in the ‘30’s. It was used to wrap everything imaginable. It has been supplanted
by plastic films today. Ordinary plastic doesn’t have any effect on the polarization of
light. When he holds the filter on the cellophane (which is on the projector), you see all
these different colors, kinda like a prism. Because of the way it is made (rolling up),
cellophane rotates the direction of polarization quite a bit. The amount of rotation you get
depends sensitively on the color of the light. As he rotates the polarized film above the
projector the color changes on the cellophane.
He then puts cellophane between two plastic sheets. Polarized light enters from the
projector. When he puts the second polarizer down we see remarkable colors, that were
just transparent a second ago. As he rotates the upper polarizer, the colors become
complementary. Yellow turns to blue, and blue turns to yellow. Because of the way
that the rotation of polarization changes with color, these are complex colors that come
out. They are not typically simple, pure colors.
We’ll leave the subject of light polarization for the time being.
There is a certain rationale to the collection of the topics discussed in the term so far.
1. Light Refraction and Reflection
a. This tells us about everything we want to know about lenses and optical
instruments, mirrors, etc
2. Color
a. Light rays exist of all different colors
b. The interesting subject of color
c. Polarization
Now we have to face the fact there are a lot of phenomena of light that we all are familiar
with, that have no explanation to us in terms of the “ray” picture.
Historically, some of the first phenomena were noticed by a Jesuit priest, Francesco
Grimaldi of Florence (1613-1663), who made his observations secretly. He noticed that
there is something strange about the nature of shadows as you examine them closely.
For most shadows, the light comes from a distributed source, and so the edges are almost
always a little fuzzy, because the light comes from various angles. He had the notion of
examining the shadows of objects with a filter, when the illumination came from a bright
pinhole – a single point of light. He found that there are color fringes in the shadows of
objects. If he filtered the light, these were alternately light and dark fringes.
He also noticed that light always seemed to spread out as he added other holes to his
experiments. He called this “diffraction.”
Demonstration: Instead of using pinhole illumination we are using a red laser (and a razor
blade). We see that the edges around the razorblade have fringes, wavelike structures. We
have different colors for different fringes. Patterns are always much simpler when we are
dealing with monochromatic light.
We can never explain this using the concept of rays.
Soap bubbles – they have wonderful coloring effects that show up in them. His assistant
Dan makes a soap film around a circular frame and shines a white light source on it.
There are “beautiful colored bands” which get closer and closer together toward the
bottom of the soap bubble film (where it is much thicker).
Robert Hook was the first person to investigate soap bubble films. There is a certain
amount of hydro-dynamic flow that takes place within the film. The liquid gets thicker at
the bottom, and thinner at the top (gravity).
Another phenomenon we can observe is the colors that are in puddles. Inevitably there is
some oily material floating on the surface of the water. Dan puts a drop of kerosene on
the water. We quickly see these bands that Grimaldi talks about. Hook demonstrated that
if one is to see these colors it is necessary to have two reflections taking place (both the
upper and lower surface). The colors that we get depend on the thickness of the film
(floating on the water). If the refractive indices of the water and the oil are made to be the
same there is no 2nd refraction since there is no discontinuity in the refractive property.
When we don’t have this second reflection we don’t see the colors.
The existence of this coloring effect depends on having both reflections taking place
at the same time.
Newton studied this and used monochromatic light. Instead of colors we see light and
dark patches, therefore. He pushed together two sheets of glass, the bottom one was
perfectly flat, but the top one was convex so that it made close contact at only one point
on the sheet of glass below.
The important reflections that take place are the ones that take place from the two
different surfaces. He saw concentric light and dark rings. He used the successive radii of
these rings to show that there is something periodic in the nature of light. “Newton’s
Rings” are light/dark/light/dark/light succession. The dark rings have a succession of
radii that are proportional to the square root of 2, the square root of 4, the square root of
6, etc. And the light rings have radii proportional to the square root of 1, the square root
of 3, the square root of 5, and so on.
Lecture 3/1/06
Grimaldi & Diffraction. He was looking at the shadows of objects cast by pinhole light
sources. He found that shadows were never sharp, and that they had these strange
“fringes” around them. This is the same as the pinhole camera. If we made the pinhole
sufficiently small, then we would have an image of improved definition. Down to a
certain size of pinhole, if you make the hole any smaller, the image gets worse, and
dramatically so. This was a clear indication that the light as a ray scenario was
incomplete.
Robert Hooke discussed the colors that one saw on films. He found that the colors
depended on the thickness of the film, and that the reflections one saw were from both
the upper and lower surfaces of that film at the same time.
Newton went on to investigate the same phenomenon using curved and straight panes
imposed on one another. One pane was parabolic in shape, while the other was
rectangular. Newton saw simultaneous reflections of these two inner surfaces. See
drawing #1. This situation is analogous to the reflection of light from a thin layer of oil
on the surface of water. But it was the difference that it was air that separated the two
sheets of glass for Newton.
Newton came up with the notion that light consists of freely moving particles and that
when they entered glass they were subject to certain forces. When light is reflected it
undergoes no horizontal (x-axis) velocity change. Light is subject to no forces parallel
to the x-plane of reflection.
See drawing #2.
During refraction, light undergoes changes in the vertical vector component, but no force
is exerted in the horizontal direction. Velocity increases in the vertical component.
Newton’s explanation of refraction: Light goes faster in glass than in air.
Logical as this may sound, it is totally wrong. Newton was wrong. Thanks for spending
all this time teaching it to us, dickhead.
Newton found evidence to the contrary to the notion that light is composed of particles.
Central to this understanding is the phenomenon of Newton’s Rings. He found that the
radii of these concentric rings are periodic. Using monochromatic light he found that
there are “dark” and “bright” rings. The bright rings √3 and √5; while the dark rings √2
and √4 as they extended outward from the center of the circles. He described this periodic
phenomenon as “fits and starts” in the nature of light. There is something spatially
periodic in the structure of light.
Periodic Motions
- we start out with motions that are periodic in time, motions that repeat
themselves in time, and thus have a period
- there is always some minimum period of time in which the motion repeats
itself, often denoted as “T”
- The simplest periodic motion
o Ball at the end of a string; he swings it around his head at a constant
speed
- Then he goes on to describe the basics of converting degrees to Radians
- As we have velocity going faster, we see that theta increases. The symbol for
this is omega x theta (omega looks a lot like a w)
- Omega is a kind of angular speed.
- (see written notes for this summary of vectors, angles and speeds)
When we spin a ball around our heads on a string, this is technically accelerated motion.
Acceleration, in physics terms, is more general. When the vector is changing with time,
you have acceleration, even if its magnitude never changes.
Centripetal acceleration is when something is always being accelerated toward the
central point about which the object is rotating.
Lecture 3/3/06
(5:00)
Math: Sines and Cosines
We reached a point at which where our initial view of light as a straight line won’t go
anywhere. It won’t explain the colors you see in films, or diffraction (the bending of light
around shadows).
Newton concluded that something periodic is taking place in the propagation of light.
Now we’re going to start talking about periodic motions. Historically, the understanding
of the nature of light stood still after the death of Newton. It’s not surprising that this
hundred years was the period where the mathematics of motions was developed.
A couple of words about the dynamics of simple oscillatory motion. The simplest
periodic motion is when you have an object going around a string.
θ=ωt
Period = T
In circular motion we use radians, not degrees.
Frequency – the number of revolutions per second.
He gives us the derivations of some of the equations that are in the lecture notes.
Uniform circular motion: the angle increases uniformly with time. While the magnitude
stays the same, the direction keeps changing. This is an example of centripetal
acceleration because the mass on the end of the string is “driving toward the center.” This
is different from the word “centrifugal force.”
He twirls a ball on a string go around his head. The ball seems like it is pulling away
from him (centrifugal). But this is not really the case, since when he swings the ball
around his head he is pulling the ball toward him. He is the origin of the centripetal force.
Because the string is taut, he feels centrifugal force, but this is mistaken.
Newton’s great contribution to mechanics was stated in his laws of motion. He said that
acceleration responds to force.
(21:00) An abstract definition of a different kind of motion: the projection of
He uses a contraption on the projector screen above that shows us the point as well as its
shadow – which oscillates back and forth uniformly. It’s a linear motion – back and forth
– that represents the projection of the circular motion.
The complete story of simple harmonic motion.
The acceleration responds to the net force (add them all up).
Demonstration: He has a neon orange mass on the end of a spring. In equilibrium it is at
rest because the downward force of gravity is equal to the upward force of the spring.
The net force is proportional to the displacement.
Newton’s law of motion: the acceleration is proportional to the displacement. He gives
the mass an initial “displacement” and it bounces up and down. It obeys the condition
that acceleration is proportional to the displacement and directed against it. Omega is the
constant of proportionality. We are satisfying Newton’s famous equation of motion when
we do this kind of simple harmonic motion.
(30:00) He goes into the basic derivations of trig: sines and cosines to show what happens
with angles larger than 360 degrees.
Demonstration: he has a ball on the end of a spring, and a nearby uniformly rotating disc
with a ball attached to it as well. There is a projector in front of both of them. We can see
the two balls oscillating up and down in harmony when displaced just right. But they get
out of step.
Demonstration: then he shows us that simple harmonic motion can be horizontal as well.
It’s just one slider on an air track with two springs attached to either side. It’s just
difficult to arrange because of friction. So we get rid of the friction by using and air track.
By forcing air out of the aluminum beam, the sliders get lifted onto a cushion of flowing
air, which makes their motion relatively friction-less.
Another example of simple harmonic motion: not perfect because it’s not taking place in
a straight line. We have two pendula that are hanging, but are connected to each other by
a spring. (it’s not perfectly accurate, because the motion is taking place in a sort of
circular arc). There is a combined effect of gravity in restoring the pendulum, plus the
spring which is acting on both pendula.
Then he shows us how to set up the procedure so that one pendulum starts with PE, but
the other is stationary. Eventually, the one he started in motion comes to rest while the
other one begins to move back and forth.
If he gives them equal initial displacements, then the spring that connects them never gets
stretched/stressed, and then both of them oscillate in unison. Simple harmonic motion
using two mass points.
All oscillating systems have what is called “normal modes.” It’s a mistaken name,
because they are really rather abnormal. The notion is that normal modes are simple
modes but they are not modes that you would excite in any normal excitation.
(42:00) “Beating”. Both motions are taking place at the same time, and they are being
added up algebraically. They have two different frequencies.
(45:00) The algebraic sum: mathematically adding up symmetric and asymmetric
oscillations.
(50:00) We have a pair of masses on the air track. He displaces one while holding the
other down. We have the two normal modes.
3/6/06 Lecture
Last time we spoke about simple harmonic motion: oscillations. We have defined it as the
projection, on a straight line, of uniform circular motion.
Then we moved onto more complex systems. In general we are talking about elastic
responses: ones in which the force that tends to restore the system to its equilibrium
position is proportional to the amount of displacement away from the equilibrium
position.
In a pendulum you don’t see elasticity per se, but rather the restoring force of gravity. So
a simple pendulum, hanging by itself, is a harmonic oscillator.
Then we got the example of the two pendula hanging side by side, connected by a spring.
The point of that was to show us what happens when we couple two such oscillators
together. There are still harmonic oscillations going on, but also in a sense they are
masked.
- it has 2 simple ways of oscillating, they’re called normal modes of oscillations
o Symmetrical model: both pendula have identical frequencies, they just
oscillate together doing the same thing, the spring connecting them
never gets stretched or compressed – identical motions
o Asymmetrical mode: equal and opposite initial displacements; the
weak spring connecting the two pendula does get stretched and
compressed, which raises the frequency of that oscillation, so that this
is a perfectly harmonic oscillation, but of higher frequency. Then we
got to the idea of the algebraic sum of those separate species of
motion. They are two periodic motions, but they have different
periods. It looks rather irregular and puzzling – the spring connecting
the pendula is so weak that the two frequencies of oscillation are
nearly the same. As they get out of step they tend to cancel each other
out.
We have a couple ways of illustrating this – we’re going to use projection and circular
motion.
15:00 demonstration: he has two platters, one on top of the other. There is a ball on the
top platter, and using a projector we see that ball’s motion as a horizontal projection on
the wall. We see how the amplitude increases and decreases.
If you’re going to sum up two trigonometric arguments (cosines) that increase uniformly
with time you’ll get this “beating” effect.
Normal Modes of Oscillation: last time we used first a single slider, and then two air
sliders on a track to demonstrate normal modes.
(19:00) Now we use three sliders. He starts them out in a random way – we don’t see any
regularity. We see “random hesitations” and “funny behavior;” it’s irregular and chaotic.
With only three sliders there is a lot of complexity. He’s trying to set up the right
conditions for demonstrating normal modes.
(20:40) We see a fair approximation to a simple periodic/harmonic motion. The
amplitudes of motion are somewhat different. The central one is moving further than the
ones at either side – by a factor of the square root of 2 (not a random number).
Then he gets the two on either side of the central slider to move in asymmetrical motion
(their directions of motion are opposed to one another).
As a general rule of thumb: When you have n masses connected together elastically, they
will have n different normal modes of motion.
So we have three masses on the air track. He gives them a “knock,” and we have all three
kinds of modes.
The reason any motion of these three objects can be represented like this is that these
three normal modes represent three fundamental ways the system can oscillate, and you
have just enough variables to describe any motion. There is a certain simplicity that
underlies the behavior of all oscillating systems (like musical instruments, for example).
(28:00) Another demonstration: a computer program is projected onto the screen above
the class. We were just witnessing longitudinal oscillations – back and forth. Now we’re
going to see a different situation where we have a rubber string with a succession of
regularly placed beads. The distance between the beads is the same. We have an elastic
string of beads.
We see examples of symmetric and asymmetric oscillations.
(32:00) Now he uses three beads. We see transverse oscillation (that just means it’s
perpendicular displacement). What’s going on here: this is not a photograph of nature.
(34:00) Now it’s seven beads on the computer program. We see what looks like
organized waves traveling across the system. But all in all we’d be pretty hard-pressed to
find some regularity. On the other hand, there are normal modes in the system.
Alright, the normal modes are pretty damn complex. They are defined dynamically: any
motion looked at over a finite length of time can be regarded as a sum of periodic
motions, provided you are willing to restrict your attention to that finite length of time.
Normal modes have frequencies that are fixed dynamically.
(43:00) We are about to embark on our preoccupation of waves. We are moving toward
an understanding of light and sound as comprised of waves (earthquakes, too). So what is
a wave?
The simplest system that we can imagine is a long, stretched rope. Let’s assume that the
rope is semi-infinite in length. So we hold one end of the rope and “snap” it, giving it a
certain displacement in time. It will create a sort of bulge in the rope, a transverse
displacement because it’s perpendicular to the length of the rope. The speed is “c.”
The rope stays still until the excitation begins to move. You move one interval of rope, it
will move/displace the next interval on the rope. The result is that if you started out by
moving the rope up and down, this upward state of motion will be transmitted along the
rope. Whatever it is, it always moves with the same speed (c). At a later moment, the
wave will assume the same sort of configuration, except it will be the opposite.
(48:00)
3/10
Waves on a rope – waves in more dimensions than one. When we have waves of the
same frequency going in opposite directions on a string. Addition theory – add up to
opposite waves.
Waves are harmonic – oscillating with a fixed frequency
Add up two waves of same amplitude + frequency, get the product of two sine functions,
oscillate independently of one another
Nodes – when x=0, no displacement on string, everything cancels
No constraint on the frequency or value of the wavelength
Wavelength: distance wave goes in one period
Low frequency means large wavelength
Semi-infinite string – fixed at one end
Points of maximum excursion – anti nodes
What kind of vibrations on fixed (finite) length?
Neither end of strength of string can be moved. What are the normal modes of
oscillation?
Fundamental frequency: simplest sort of vibration you can have
When you fit three half wavelengths, called third harmonic
Stretched string of finite length – can have infinite sequence of normal mode oscillations
What happens if you pluck string?
Depends where you pluck it
All music instruments make sound that are superpositions
Longitudinal – vibrations back and forth along the same dimension of wave
What about in 3 dimensions?
Solid materials, like steel, very elastic but rigid, so transmits quite rapidly motion of one
element to the next. So transverse waves travel rapidly in steel. Longitudinal even faster.
Fluids don’t like to transmit transverse waves at all – air no transverse wave, but
compressible waves
How to generate sound?
Loudspeaker experiment. Turn frequency up begin to hear sound
Oscilloscope shows variations of air pressure as a function of time
Larger loudspeakers produce lower sounds
Guitar
Tighter the string, the faster the speed of vibration
3/13
Integer multiples known as harmonics
In a violin – you don’t hear string, you hear oscillation of wooden body of the violin
(sounding board)
Tuning fork another oscillating object
Produces purely sine or cosine wave – that’s why it’s sound is too pure
Oscillations of tuning fork are forcing oscillations of piece of wood
So musical objects: oscillations communicated to something else
Hit two together – beats – there’s a throbbing
Throbbing (beating) is slower when frequencies closer together
One of higher frequency will sooner or later get once cycle ahead – this time is known as
periods of the beats
Beat frequency – difference of the two frequencies (absolute value)
Sound waves are longitudinal waves not transverse
Superpose – add up harmonics (ear superposes)
Pipe
Open at both ends – pressure going to be atmosphere pressure
Sound pressure=0 at the two ends, but can have higher pressure in the middle since the
pipe holds it, but it extra pressure must =0 at both ends
Closed at both ends
Can have accumulation of pressure or deficits of pressure at the two ends
Air can’t move at either end so must either a maximum or minimum at the closed ends
Closed at one end, open at the other
Pressure max, min at closed end. 0 at open end
You get only the odd harmonics
Water Waves.
1. Are not as simple as sound or light waves because they have a great deal of
dispersion (different wavelengths travel at different speeds). So we have to
idealize their behavior in the models that we use.
2. “Bow Waves” do not have a perfectly sharp wavefront.
V
V>C
The boat is moving with
velocity V, which is
faster than the speed of
the waves
Shock Waves
1. They are very sharply defined, as opposed to water waves.
2. They are very dangerous, but not because of anything to do with air pressure: if
air pressure is 14 psi then the disruption of the shock wave would only be 1/100th
of that: .14 psi. A supersonic airplane will produce shockwaves that exert pressure
on windows. Windows have quite a few square inches to them, so although .14
psi sounds very small, it is a big deal for windows. Supersonic commercial
aircraft were
Supersonic shock wave
generated by supersonic
planes
looks like this: conical
shape
If you were a person at point o below, you
would hear the boom equally in both ears.
Which way would you look to find the
plane? You would look to the left at first,
but the plane would actually be to your
right by the time you had looked
3. Huygens: considered optical laws. Demonstration: he uses a bluish film to make
these so-called plane waves. The waves look like ripples in a pool. It shows wave
reflection.
Waves bump into
each other to form
patterns called
“reflections.”
He then also tries to procure refraction, by varying the speed of the waves propagated in
this demonstration. This causes the wavefront to get bent.
Thomas Young
1. Advanced the study of light waves after Huygens. Was the quintessential child
prodigy. Started building microscopes and telescopes after boarding school. Was
fascinated by near-East languages, and then began teaching himself mathematics.
At about age 19 he went to medical school where he dissected the eye to find out
what focuses the eye. He was the first to understand astigmatism. By 21 he was
admitted to the Royal Society of London (natural philosophy).
2. He made a pronouncement to the Royal Society: that there was a relationship
between strings and Newton’s rings.
3. (35:00) Newton’s rings recap: you have two sheets of glass, the upper one planoconvex the bottom one flat and they are pressed together. White light is shining
from above. You look down at it and you see this succession of colored rings. If
you look at these rings in one color (not white) you see alternating light and dark
rings. (Hook had already told us that we see similar effects of oily substance on
water when light is reflected from those two surfaces.) A light wave is being
reflected into your eye from the top surface as well as the bottom. These double
light waves oppose one another. When they meet the eye they cancel one another.
Periodically you get cancellation and re-enforcement.
There are two
light waves,
reflections,
destructive
interference
occurs when dist
=¼λ
4. Because he knew calculus he was able to find λ (red light) = 1/36,000 in. or 0.7μ
(micrometers 10 to the –6) and λ (blue light) = 1/60,000 in. or .4 μ. These
measurements are good to this day.
5. No one believed him.
6. Wave theory and interference: he said that with respect to Newton’s rings, these
waves were interfering with each other. Today we would use other words to talk
about this phenomenon. These waves are just linear super-position; adding them
up algebraically.
7. The succession of Newton’s rings corresponds to increasing distance between the
two glass plates. His notion was basically that you can add light to light and get
darkness. And this was not acceptable to his friends at the Royal Society.
8. He also had a theory of color perception: red, green and blue receptors are in our
eyes, and varying stimulations of these produces different colors that we perceive
(like a TV).
4/3/06 Science A29 Lecture
We’ve been talking about Thomas Young, who did more to shape our notions about light
than anyone in the 19th century. He began to think in terms of analogies between sound
and light.
His first thoughts were relating the structure of organ pipes to the nature of light. He also
researched Newton’s Rings.
Young understood that given the spacing between the two sheets of glass, there would be
two reflections. The top sheet of glass and the bottom sheet of glass each reflect a light
wave. When the two sheets are separated by a quarter wavelength you get complete
cancellation. The dark rings would in Newton’s Rings experiment would be an example
of this. Half a wavelength leads to reinforcement.
Thomas young was able to figure out the distances between the two sheets of glass, and
determine thereby the characteristic distance. This was the distance between the two
sheets of glass that corresponds to the light and dark rings. Cancellation/destructive
interference/superposition, these are all very similar words.
The intervals in the distance of the dark rings would correspond to the distance between
the glass plates.
*Every time the distance between the glass plates is a quarter wavelength you get
cancellation; every time the distance between the glass plates is a half wavelength you get
reinforcement. This discovery leads to the notion that he could measure the wavelengths
of light.
He had a bit of difficulty presenting his theory to the Royal Society simply because no
one would believe that you could add light to light and get darkness. There was also
another problem here – that the waves should have gone the same distance in the center,
and so you get a dark spot. It didn’t take him long to guess, but that in one of the two
reflections there was a reversal of phase that lead to the cancellation. But he didn’t know
where the reversal of phase took place – at the upper or lower surface. At this point of
contact you get non-reflection, and the only way to explain this is reversal of phase to get
cancellation.
His colleagues probably didn’t buy his conclusions. So he was left trying to find other
ways to prove that light consists of waves. He was in a sense taken Huygens’
suggestions. Huygens didn’t really talk about periodic nature of light. There is another
relationship between that of sound and that of music and that of color, they are all
periodic. I don’t know what the point of this is.
In 1801 ultraviolet light was discovered. Young was aware of this. It wasn’t easy to
detect uv light. Young actually showed the existence of Newton’s rings in ultraviolet, and
showed that the wavelength had to be shorter in ultraviolet than it would have to be in
blue light.
Later on he began investigating Grimaldi’s fringes. Remember that Grimaldi described
all these strange shadows that appeared around objects when using light that had first
gone through a pinhole. In monochromatic light this produces fringes. It’s hard to see,
though. We used light from a laser beam to show them to us.
Young did an incredibly ingenious experiment for his time: he took two types of glass
with different refractive indices. He was able to find a medium that has an intermediate
refractive index. So that light passing through this system would go through three
separate media, with successively increasing indices of refraction as the wave got to the
bottom plate (this was oil of sassafras). When he did this, he got a bright spot at the
middle.
(27:00_ His examinations of the fringes, also see handnotes.
The two pinholes become coherent radiators of light: oscillation-for-oscillation the waves
emitted by these apertures are identical. That’s the point of using pinholes; all parts of the
wave are operating in synchronism. It sets up the preconditions to observe the fringes.
Coherent = waves are oscillating in perfect synchronism
(38:00) we can see coherence in ripple tanks, which we see oscillators that vibrate in
unison. You can draw the perpendicular bisector between the two sources to get a line of
points that are equi-distant from the sources. You can see where they reinforce one
another, and when you get cancellation. You get patterns of alternating reinforcement and
cancellation.
He then takes two speakers that are on a plank, they radiate with each other step by step.
He rotates the axis of the speakers, you can hear the intensity of sound go up and down.
This is due to reinforcement and cancellation. The audience finds themselves in the fringe
pattern, and thus hear different notes.
(48:00) Then we see the ripple tank – reinforcement and cancellation are graphically
depicted. Destructive interference,
He then sets up the Young demonstration using a laser (because it’s coherent you get the
same wave oscillating from the two slits. We see these red worm-like thingies moving in
a line. Finer spacing in the slits tends to spread the pattern out.
Lecture 4/7/06
We’ve been discussing an event that’s the turning point in the history of the
understanding of light: Thomas Young first had a theory about the origin of Newton’s
Rings. He then experimented with Grimaldi’s Fringes (fringes that are only seen using
pinhole illumination). To understand a little better what was going on with Grimaldi’s
fringes and the apparent fact that light seemed to go in both directions around obstacles,
Young did his double pinhole experiment.
The idea is that a wave arrives as a uniform wave, makes the pinholes act as two separate
sources of waves, which produces interference. This produces re-enforcement and
cancellation. This leads to a succession of light and dark fringes.
Suppose for instance one didn’t use pinholes, and instead used a simple lightbulb or
flashlight – would one see the same fringes?
The word coherence is used most often in this connection. You have to have a wave
falling on these two very closely spaced pinholes. You have to create a situation in which
these two pinholes become identical sources. The wave is doing identical things from the
two different sources. As soon as you have waves coming from different directions, this
ceases to be true.
As soon as you have a source of finite size, the waves are going to be coming from
different directions, which means that there will be a mixture of waves from different
directions arriving at the pinholes. These mixtures don’t behave identically. This is the
very essence of the coherence, which is of utmost importance in our discussion.
It was very important that Grimaldi had the brilliance of using a pinhole to produce
the effect of one spherical wave falling on the pinholes.
We have been using lasers in class, which is very important for these demonstrations. The
light is bright, and it’s also monochromatic (color is very well defined). A laser produces
a single wave (plane wave), that’s coherent.
Where do we see the fringes, and how do we see them?
(20:00) See hand notes
Young worked out what the wavelengths of red and blue light were. We can ask what the
corresponding frequencies of oscillation are: we found that the frequencies of these two
wavelengths of light are really incredibly large. It was clear to Young that there was
nothing in our bodies that responds to things on this order of magnitude (or minitude).
If you take any interval of time (millisecond for example, is enough for a million million
vibrations). Young assumed that it was the square of the displacement in the field. It was
a reasonable guess that we are seeing the square of the displacement of light.
(36:00) Can you calculate the intensity pattern that Young saw?
(49:00) Demonstration
If you align the two parallel slits on the cardboards handed out at the beginning of the
class you’ll see Young’s pattern. Angular scales are different for different colors.
Diffraction/Interference phenomenon.
Young also had problems with presenting this to the Royal Society. He was roundly
denounced by his colleagues, even though the professor thinks this was the most brilliant
analysis of the nature of light.
4/7/06 A29 Lecture
We have come to the point where people of the 19th century may have actually realized
that light was indeed composed of waves. Young was given a very hard time, even
though from a modern standpoint his demonstrations were brilliant and persuasive. It
took quite a bit to persuade the hard-headed British. Young didn’t succeed in doing it. He
printed up a pamphlet to respond the criticisms of his work, but very few copies were
distributed or sold.
Young’s numbers for the frequencies of red and blue light were some of the largest found
values for them for his day and age. Interestingly, they don’t even span an octave. To
span an octave, there would have to be a factor of two in the frequencies. One of the
lessons we’re going to learn is that this is just the tiniest crack in the spectrum of what we
call light, even though most of it is invisible to us.
Around 1805, after having been fired from the Royal Institution in London, probably
because he wasn’t a terribly persuasive lecturer, he gave up his work on light. Other areas
that worked on included: first systematic studies of elasticity, capillary action (he was a
doctor after all), blood circulation, he developed a theory of the tides, pioneer of lifeexpectancy tables (actuarial work), and lastly his fantastic command of Oriental
languages enabled him to translate the middle script of the Rosetta Stone and parts of the
hieroglyphs. He went on thinking about light some, and engaged in correspondence with
a Frenchman, Fresnel.
Fresnel (1788-1827). As a civil engineer he was rather inquisitive about the things he saw
in nature, but he had no time to indulge his interests. He was a Royalist, and he joined the
forces that opposed the return of Napoleon from Elba. When the Royalists were defeated,
he was placed under house arrest. With nothing to do, he busied himself conducting
experiments much like Young. From 1808 on he was fairly active and repeated some of
the things that Young had done. One of his first innovations was to use slits instead of
pinholes.
Slits give you the same geometry, but rather more light, so it’s easier to see the fringes.
His first observations were really to repeat the observations made by Young. He saw
Young’s interference pattern. But then he went further by asking what happens if you
have two or more slits?
See hand drawn notes on Young’s double slit experiment.
Suppose we have more than one slit. Suppose that we have N slits, we don’t really care
how many in all. The condition for re-enforcement will be exactly the same as when you
have two slits. What the equation for the “n-slit reinforcement condition” is saying is that
we have constructive interference when the delta r is equal to one, two, three, four, etc,
etc wavelengths
The condition that leads to the reinforcement maxima is always the same regardless of
the number of slits that you have. That means the amplitude of the wave is made N times
what you’d get from a single slit, but the intensity is N-squared. The essence of this
addition is that you get a very large build up of this reinforcement process.
What happens when you do it with two slits, three slits, four slits, etc, etc?
Check the hand written notes for the math.
(30:00) He puts the graphs of the intensities of various numbers of slits. Although the
principal maxima remain at more or less the same place, there are subsidiary maxima in
between (# of subsidiary maxima is N-2). The important thing to note is that the maxima
are at the same angles, but they become narrower as N becomes larger. As N becomes
very large you don’t even see these subsidiary maxima at all any more, and what you see
is very narrow diffraction peaks with a width inversely proportional to N.
(34:00) With a great many parallel slits (1000’s of them), you get diffraction along the
wavelength of light. The maxima are different according to the wavelengths of light.
These correspond to the colors of the spectrum. Light spreads according to the
wavelength. So diffraction gratings spread light out into spectra. This is excellent for
spectroanalysis. It gives a formula for the deflection angle given the wavelength of light.
Corresponding to different wavelengths of light, these maxima will lie at different angles.
He then goes on to the demonstration: we see a red laser beam projected on the above
screen. 2 subsidiary maxima are then seen, which tells us that there are four slits.
There is the overall modulation of the single slit pattern as the number of slits changes.
Diffraction Gratings:
It was hard for scientists that came before us to produce diffraction gratings with a great
degree of accuracy.
Demonstration:
He turns on a lamp, and then instructs the class to hold up diffraction gratings to see the
first and second order spectra. As you try to see greater orders, you have to look more
and more to the side, which eventually becomes impossible.
Then he turns on the lamp below and asks the audience what they see. You see the
characteristic spectral frequencies emitted by a hydrogen lamp.
4/10/06 A29 Lecture
We’ve been talking about Fernell’s investigations of diffraction and interference. And
one of the things he did was to investigate the interference patterns created when you
have N parallel slits (diffraction grating) with a constant spacing d between them. And
we have derived a formula for the intensity.
- Using this setup we can see a succession of maxima
- These maxima are very useful for spectroscopy because they lie at perfectly
predictable angles given the wavelength
- Glass prisms also spread light into a spectrum, but the dispersion of glass is
not something we can control so easily because it requires an awful lot of
calibration. It’s much harder to measure wavelength.
- So diffraction gratings are quantitatively very useful. And to the extent to
which you can make those spacings very tiny, you can achieve a great deal of
spreading out of the spectrum they became very accurate devices for
measuring wavelengths.
We are now going to discuss a situation in which we allow the number of parallel slits
become infinite. But first we need to talk about various types of spectroscopy and of
diffraction gratings.
Reflection Grating Demonstration
He then turns on the projector because it provides a convenient spotlight. In his hand he
has what looks like an ordinary mirror. Actually, it’s a sheet of flat glass that has these
diffraction rulings directly etched on the glass and a coating of aluminum that has been
evaporated over it. What you have is not a transmission grating, but a reflection grating
(which is essentially the same thing). He turns out the light. We see the first order
spectrum of light to either side on the projector, and a little bit of the second order
spectrum. As you let the spacing between the slits go to zero these spectra/interference
maxima move out to larger and larger angles. Presently they get out to 90 degrees and
then you don’t see them any more. Since the wavelength of red light is almost twice that
of blue or violet light the angle theta is correspondingly larger.
The following demonstration is useful for two things: illustrating diffraction and its use in
forming diffraction gratings, and the other is spectroscopy on the hoof. What is the
application of these things?
- These diffraction phenomena have made available all manner of wonderful
wrapping paper. What these things are, is really reflection gratings. The
people that make this stuff for a living have found out that it’s a good idea to
put various patterns in their slit-work
- You also have the recording that’s carried out on compact discs. How does
that work? The variation in the reflection of light is what produces the sound
as the machine reads the CD. Because the coded tracks are close together
The Demonstration
He then turns on what looks to be a column of tubes, each emanating a different spectrum
of light. On the top is yellow, then purple, then mauve, and then red at the bottom. The
purpose of this demonstration I gather is to show spectra emission lines.
If you look at a florescent tube you’ll see that it consists of spectrum lines that correspond
the mixture of gases that are present in that tube. If you look hard enough you’ll see
second and even third order spectra.
Understanding spectra was the skeleton key for unlocking the secrets of atomic structure.
What about dealing with slits of finite length (up until this point we have assumed that
the slits have had no width for all practical purposes). In order to deal with actual
situations in optics, we have to account for the fact that these slits have a finite width.
That width is almost always large compared with the wavelength. One tries to make the
slits as narrow as possible, but the wavelength of light is awfully small.
This part of the demonstration had to do with diffraction gratings: a multi-slit sheet of
plastic that spreads the colors of the incident light. There were four different light sources
(incandescent, hydrogen, helium, and neon). For the incandescent source, each order of
the interference maxima looked like a rainbow spectrum of colors, but for the other
sources the maxima were just discrete emission lines.
The derivation of the following part is really tricky, so perhaps this might need to be
revisited at a later date, maybe with Shaun.
(35:00)
So we’re going to consider a limiting case. We’re going to deal with a slit of finite width.
One way to conceptually do this is to look at our situation (see hand notes) in terms of the
slit being composed of a vast number of slits very close together. So we’re going to use a
limiting process, in which we let the number of slits, N, go to infinity, and the distance
between these slits go to zero. And then we’re going to add up the intensity that results in
effect from that infinite number of slits.
The first thing to realize is that as you let the spacing of the slits become smaller and
smaller, the angle at which the maxima lie will increase. The wavelength of course is
fixed. As d goes to zero, the sin of the angle is going to increase. But the sin of the angle
can never be larger than one. So what’s going to happen as we let the spacing of the slits
go to zero, is that as soon as the spacing (d) becomes equal to the wavelength, then we’ll
find that even the first order maxima has moved all the way out to 90 degrees. Letting d
go to zero means all the higher order maxima tend to disappear, and you are left only
with the one in the center.
d sinθ = m λ
N (the number of slits) goes to zero.
We are looking for the shape of the diffraction pattern that emerges when we regard a
finite width slit as made up of essentially infinite number of very closely spaced slits.
Check hand notes at this point.
Ordinarily the ratio of the wavelength to the slit width is a small number. And that means
then that this angle theta is small. So typically these diffraction patterns for a single slit
will be cast over a very narrow angular range. You can see that it must have zeros.
So there is a succession of angles at which the single slit diffraction pattern has zeros.
There is something very
Any time a light wave goes through a light wave of width a, there is a certain spreading
out that takes place, and a certain fuzzing out of the angle at which the ray is thrown.
Instead of the light being thrown uniquely in the forward direction, it’s thrown over a
certain range of angles. And that range of angles is characteristically the ratio of the
wavelength to the size of the aperture. This is an extremely fundamental observation for
optics. Extremely important. The spreading that takes place is thoroughly characteristic
of every sort of optical instrument.
(49:17) When light goes through a circular aperture you get a similar result.
Airy, the astronomer. There is a limit to the resolving power of any telescope or optical
instrument. The Rayleigh Criterion: The smallest angle that one can resolve = 1.22
λ/2R; R is the diameter of the lens you are looking through.
4/12/06 Lecture
We’re coming to a kind of turning point in the course.
What has happened up to this point has been a great deal of consideration of the general
properties of waves. Then we had the demonstration that light consists of waves. Just
what kind of waves light is/are, remained in question during the 19th century.
We are now looking to understand what those waves are made of.
He first wants to conclude the story of Fresnel’s work and how it led to the diffraction
grating. And, then there is one more very important problem.
In our discussion of Young’s experiment that had two pinholes, to our dialogue about
Fresnel generalizations involving slits, he assumed implicitly that those slits should have
zero width.
You can’t have a slit with zero width
1. if you gave it zero width no light would get through
2. you can’t do anything mechanically with knives, razorblades that doesn’t create a
slit that is considerably larger than the wavelength of light, which is awfully small
This does have a considerable bearing on the distribution of light that has passed through
the slit. He has mentioned already that in Young’s interference pattern, we see a certain
modulation in the intensity graphs even though we predicted it to be more of a stable
wiggling.
One of the things we do when we consider slits of finite length is explaining that
modulation. The most important element is that “any time that light passes through an
aperture of any finite width compared with the wavelength, there is a certain tendency
toward spreading. This spreading is generally called diffraction (term introduced by
Grimaldi). Diffraction is one of the things that waves simply do.
One of the ways we approach a slit of finite width is to use our formula for the intensity
produced by a diffraction grating, and let the separation of those slits go to zero and let
the number of such slits become infinite. In that limiting process we end up with in effect
amounts to letting light go through a single slit of finite width.
(10:00) Single slit
Inasmuch as the wavelength is always small compared to the slit width (using everyday
tools), the angle theta is always small. The angle is the ratio of the wavelength to the slit
width.
Light is passing through a slit of width a, and it gets spread out into this characteristic
pattern and we observe this in the graph on the hand-written notes. We can also see the
diffraction patterns. This is the pattern for a straight, single slit.
(14:29) When you use a circular slit, you have to use. This was first seen by G.B. Airy
1835. He was looking under very high magnification at the image of a star in telescope.
It’s not a perfect point (because they’re not). Stars are so far away that the ratio of their
diameters to their distance is even smaller than this tiny angle. Let’s say you have a very
powerful telescope, could you see the star as a disc?
The point is that a telescope necessarily produces an image that looks like CA 1 (hand
notes), instead of being a perfect point. If we had a perfect point source, CA 1 would be
the graph of the intensity of the image produced in the telescope. It has a certain angular
width by the diameter of the telescope lens.
Now all our demonstrations have dealt with slits that were perhaps 1/10 or 2/10’s of a
mm apart. They gave patterns that we could project onto the screen that have a certain
finite angle associated with them. You don’t think of a telescope with a lens of a meter in
diameter (refracting telescopes). You don’t think of those as providing diffraction
patterns, but they did, and that was what was noticed by Airy in making measurements of
this sort.
Suppose you had perfect point stars in the sky, could you see them as a double star? You
would see them as two diffraction images. If the stars were too close together, the
resulting intensity pattern would not indicate the existence of two separate stars. Lord
Rayleigh calculated how far away stars would need to be in order to be seen as distinct.
That’s about the finest angular resolution that you can achieve with a telescope lens of a
given diameter.
Very good slides on this whole phenomenon.
(21:00) He shows us some slides to illustrate this thing. The larger the slit the more
forward projection you have. When the slit becomes tiny (relative to the wavelength of
light) you get circular propagation. You can also see this in ripple tanks. When the
wavelength is smaller than the slit width you get forward projection, and not an awful lot
of angular deflection.
Then he shows us the Young single slit pattern again, you see that Young’s pattern is
modulated.by the intensities characteristic of the single slit pattern because each of the
members of the pair of slits has the width characteristic of the single slit pattern.
We also see how the single slit pattern depends on color. It is spread out over a larger
angular range for the red than it is for the blue, due to the difference in wavelengths.
(22:29) Then we see the image of the Airy disk, a perfect star through a telescope. He
also shows us photographed versions of the same thing. Long Exposures expose the high
peak in the center to show us the faint rings that surround the Airy disk. We also see what
happens as you expand the size of the slits.
Demonstration: he is going to show us how resolution works by shining a double-source
(as if this were a double star), we’re shining the light from the double star through a slit
of various width. If the slit is opened up you see the two beams very clearly, however if
we close the slit the two beams merge into one eventually as we close the slit. We see
more of a modulated diffraction pattern. As we close the slit further, less light gets
through, and we have no possibility of determining that this is a double source.
(25:00) The quality of the lens is beside the point. It only has to do with the fact that light
goes through a finite sized aperture. Even if that aperture is several meters in diameter (as
it is with the largest reflecting telescopes built these days) the simple fact that a wave is
confined to that distance means that it is no longer a perfect plane wave, and has to
involve a certain divergence in the directions of propagation.
Another demonstration: how much can you possibly see with a pinhole camera? (See
handnotes) If you’re observing the image produces at the back of the pinhole camera you
aren’t going to get a very sharp image because rays will go that point over a whole range
of propagation angles. Light from all over the object will come to the same point in the
final screen. So that is a kind of angular definition associated with the ray picture.
So what kind of an angle is theta? If a is the width of the pinhole, and l is the length of
the camera, then that lack of definition is the ratio of the pinhole width to the length of
the camera. We try to improve on this picture by making smaller pinholes. Small pinholes
produce sharper images.
But, as experts on the wave theory of light, we see that this generates a problem: because
you always have diffraction by this pinhole, and that diffraction has nothing to do with
the ray picture of light, just the fact that light is a wave. That angle of diffraction, that
characteristic angle of diffraction that we saw in the Airy disk and in the single slit
problem, is in the order of magnitude of the ratio of the wavelength to the size of the
aperture. By making the aperture small, we make this ratio worse.
So we have two kinds of effects that are present simultaneously. He adds them together,
those two angles of uncertainty (see graph). Sum of the angles relative to the size of the
pinhole. Ray increases linearly with a. Diffraction uncertainty decreases as you increase
a. Diffraction is improved by using a larger aperture; uncertainty is reduced.
(31:00) Suppose we say these two uncertainties are equal to one another, that is to say we
are trying to find the minimum, by just guessing, but that’s at the point at which those
two uncertainties are equal. That then says the ray uncertainty should be equal to the
diffraction uncertainty. This is an equation that we can work with.
Using these formulas we can get the best estimation of the dimensions of a pinhole
camera. This isn’t a terribly serious answer. It just tells you the best sort of pinhole
camera you are going to get will have pinhole diameter of just tenths of a millimeter.
This raises some interesting questions:
1. How good of a microscope can you build?
2. How small are the things you could hope to see?
Anthrax bacillus is of the same size as the wavelengths of light. We would love to see
staph, human chromosome, smallpox, yellow fever etc. Can we devise a microscope that
will allow us to see these phenomena.
(36:00) We will now then turn to talk about microscopy. What is it that one is doing with
a microscope, the fundamental element in the microscope is the objective lens. It’s a tiny
magnifying glass which projects an image that you then examine through the eye piece.
Let’s say that we have an object of size x. It will then take up an angular size (measured
in radians) in the ratio of x/l. Our lens always has an aperture of size a. Because it has
that aperture it will produce the fuzziness associated with diffraction. That fuzziness will
have a size, which is the ratio of lambda to a. The upshot is that while we would like to
produce a perfect image point for point, we can’t do it. Any point on the object is going
to produce a diffraction pattern. There is a limitation to the size of lens you can build. It’s
virtually impossible to make a lens with a focal length smaller than the aperture.
We thus use blue light for microscopy, which makes it easier for us to see smaller things,
since we can’t see smaller than the wavelength of light being used.
For things smaller than the wavelength of visible light, we don’t use light at all. We’re
forced to use something that has so gross a limitation. We have made progress with
electron beams. Electron beams also have diffraction limitations. We can go down a
factor of ten by using electron microscopy.
Demonstration: we see tiny blue plastic spheres suspended in water. They are jittering
like crazy. It’s an example of something discovered by a biologist, George Brown. There
is a question of what the hell is going on with this Brownian motion, as it has been come
to be called? What are those collisions. It’s due to thermal fluctuation. What we should
see is that these are individual diffraction patterns that we are looking at. These are not
bacteria or living things, although a lot of people thought that Brownian motion was a
life-force. It seems to be a tiny organic process of particles. No matter how the camera is
focused, we cannot see better than fuzzy images. We’re using white light, which confuses
the issue a little bit. Suffice it to say that there are intrinsic limitations on oscopy,
fundamental limitations on our sight to gather knowledge. We’ll see this again at the end
of the term – the uncertainty principle. When dealing with waves there are certain
rigorous inequalities that are obeyed that put limitations on the amount of knowledge you
can gain about the world by using your sight.
This is the dramatic demonstration of the correctness of Fresnel’s diffraction picture. The
French Academy every once in a while held competitions with the notion that the best of
the contemporary scientists by submitting monographs would be able to clear up
differences of opinion. 1818 the topic was light, and Fresnel submitted a monongraph
that light was a wave based on his calculations of diffraction patterns. He was able to
analyze and predict a great many of the shapes and patterns of these fringes that they
observed. The judges included Laplace and Poisson; the most prominent and
accomplished physical scientists of their day. Each of the five judges were a giant in their
field.
The judges were not believers in the wave theory of light. One of them thought perhaps
the wave theory was correct (Arago – an astronomer). Poisson simply disbelieved the
wave theory.
Arago suggested that they go to his lab, and they conducted the experiment. We see a red
Airy disk. Dan suspends a tiny ball bearing in the middle of that spot.
From Shaun’s email:
Poisson had predicted that if coherent light was incident
on a disk, and assuming that light behaved as a wave, there
would be a bright spot at the center of the circular shadow
of the disk. Such a spot was observed in Arago's laboratory.
In the lecture demo, we also saw the bright spot at the
center of the shadow of a small ball bearing that was held
in front of a laser beam. The Airy's disk has no relevance
to this demo. The reason for mentioning it during this demo
is that Dan had placed a circular aperture in front of the
laser beam, so the Airy's disk for this aperture was on the
viewing screen before the ball bearing was brought to its
position.
The demo very clearly showed the Poisson's (bright) spot in
the shadow of the ball bearing. I should add that the demo
could also be done with a small disk (e.g., a penny). The
object does not have to have a spherical shape. The only
requirement is that the shadow must have a circular symmetry
so that the light waves diffracted from the edges of the
object travel the same distance to the center of the shadow,
i.e., the diffracted waves interfere constructively at the
center of the shadow. So the key words for this demo are
diffraction and interference, and as we know these phenomena
occur only for waves.
4/14 lecture: electromagnetism
Scientists: (from earliest to latest)
 William Gilbert—earth is a giant magnet (1554-1603)
 C.F. Darfay—two types of electrical charges; later discovered positive and
negative
 Benjamin Franklin—universality of electrical phenomena; charges are conserved
and cannot be created or destroyed
 Prestley and Coulomb—electrostatic forces are inversely proportional to r2
 Galvani—investigating electric currents in animals. Thought nerves and muscles
were sparked by metals, so applied an electrical charge to animals and watched
what happened.; electrochemistry
 Volta—electrochemistry
 Davy—electrochemistry

Oersted—electrical current creates a magnetic field (electrical current flows
through wire and compasses nearby respond as if there is a magnet); connection
between electricity and magnetism
 Ampere—connection between electricity and magnetism
 Faraday—moving magnetic field through a loop carrying a current produces an
electrical field; connection between electricity and magnetism
Demonstrations:
 Big machine with spinning wheel and two small metal balls an inch apart from
each other that makes sparks (at 15:10 on video)
o Creates large voltage differences to break apart atoms ad emit
electromagnetic energy
 Flow electric current through a vertical pipe and watch compass needles in a
circle around the pipe move to the same direction
o A vertically directed electric current induces a magnetic field around the
current (right hand rule)
 Move a magnet through a metal coil—this induces a voltage
o A moving magnet creates an electric current (current varies based on the
resistance of the wire)
4/17 lecture: E&M continued
Demonstrations:
 Rub cat fur on Teflon tube and touches it to an electroscope with a pointer;
pointer needle moves when the tube touches
o Elements of charge repel one another
 Vandagraff generator: turns on generator and three balloons are attracted to the
generator, changes polarity of balloons and two are attracted and one repels
o Generator liberates electrical charge and puts it on a leather belt which
transfers it to an Al bowl where it’s held and generates lots of e charge.
 Bar magnet put in the center of dish and steel shavings put in dish, aligned.
o Metal lines up along magnetic field created by magnet
 Two parallel lines with equal and opposite currents put next to each other; they
repel each other
o Demonstrates how magnetic fields are generated by electric fields (B field
is generated surrounding the wires); changing B field always accompanied
by E field.
Scientists: (same as 4/14 lecture)
 Galvani—frog legs can be made to jump with static electricity
 Volta—developed batteries (voltaic cells)
 Davy—separated and isolated lots of chemicals
 Oersted—observed that currents go through wires
 Faraday—magnetism generates electric current
 Ampere—detected currents and B field generates E field.
Lecture 4/19/06
(4:40) As we probably all know, we’ve spent most of the term talking about light. We
have developed the studies of light in considerable detail, proving beyond all shadow of a
doubt that light is a wave phenomenon. But just what kind of waves is light … waves
made of what?
The earliest assumption was that the vacuum was some sort of vacuum (perhaps elastic)
in which these waves are elastic waves, analogous to sound waves. This medium was
called “the ether.” In its very beginning, ether was the higher atmosphere breathed only
by the gods. By this point it was the atmosphere that pervades everything.
Did these light waves consist of elastic waves in the ether? No evidence to support this.
There was a parallel collection of studies in the 19th century that climaxed around the
Civil War – studies of E & M. They were gathered together but made no particular
coherent sense at the time.
(8:00) Demonstration
Van Der Graaf Generator: device that generates high voltages and impressive
sparks/noises. Dan holds a rod with a gold spherical ball at the end above a silver dome
and we see sparks shooting up to meet it. One of the things that we showed with this
device is that when things are electrically charged they exert attractive/repulsive forces
on one another. When you get similar charges you get repulsion.
This is the kind of action that takes place in a detection device we call an electroscope
that has a little arm in it. When you charge the electroscope this pivoted arm moves up
and because of the mutual repulsion of the parts of the electroscope.
Some people that climb mountains during thunderstorms have strange electrical
experiences. Sometimes their hair stands on end. Once their hair becomes charged the
individual hairs repulse one another. That would be the action of an electroscope.
(10:00) Demonstration
He puts on a wig, and touches the Van Der Graaf Generator. It doesn’t work for the
professor, but Dan puts on a wig and it works for him.
The effect is for real, and you become a sort of human electroscope for having done that.
(13:45) Michael Faraday’s contribution: the discovery of electrical induction. There were
several electrostatic and magneto static laws known of before the time in 1821 when
Faraday did his thing. The question was, is it possible, granted that electric currents
generate magnetic fields, that a magnetic field could generate an electric current? The
answer was yes – but only a changing magnetic field.
(15:00) Faraday’s Law: as long as the magnet is moving there is current generated in the
wire, which we detect using a galvanometer. The changing magnetism (or magnetic field)
within the loop of wire is the thing that generates the current. It has to generate an electric
field, which in turn must be a force on electric charges that runs around the ring, and it is
impelling the electric charges to run around the ring. The more abstract and deeper way is
to say that a changing magnetic field (changing with time) generates an electric field.
There were four laws of Electro &Magneto Statics via James Clark Maxwell (1861-65):
1. Coulomb’s inverse square law for the forces between electric charges.
a. A great difference between magnetism and electric forces is that
electric charges are freely moveable. Once you separate the two signs
of the electric charge you can take them wherever you like. That is
never possible with magnetism. All magnets have N/S poles, but you
can never separate those poles very far, and that’s the best you can do.
b. However, to the extent that you can separate them, that you can talk
about them in the first place there is an analogous law for magnetic
fields:
2. Analogous Law for Magnetic Fields (But without Magnetic Charges)
a. There simply is no such thing as an isolated magnetic charge. We have
lots of isolated electric charges, but there are no isolated magnetic
charges.
3. Faraday’s Law of Induction
a. Prior to Faraday, the experiments were performed with unchanging
fields. With the introduction of Faraday’s Law that changed: When
there is a field B (magnetic) field that changes with time it generates
an electric field. E varies with the rate of change of B. The electric
field it generates is curiously related to the magnetic field because it
tends to be perpendicular to it.
4. Oersted and Ampere’s Law: an electric current can generate a magnetic field.
Electric currents are surrounded by magnetic fields, that how we create
electro-magnets.
(21:00) These are the four laws as Maxwell saw them, and he asked whether we can turn
these four laws into a dynamical statement of some sort about the way E&B fields behave
with time (as well as with the special variable).
We are going from statics to dynamics.
He started writing differential equations, and they looked pretty damn scary to his
contemporaries. Maxwell was convinced that there was an elastic ether underlying
everything, a medium. He also believed like Newton that everything is dynamical at its
base, and that you can write down equations of motion and solve them to predict the
motion of everything as a function of time.
He made a mechanical model for the ether. The ether, is nothing at all. He was
constructing a mathematical model of nothing at all, the complete absence of all matter.
How did he do this? The method that he used was bizarre and completely forgotten. He
divided empty space into cells, and put into each cell a kind of mechanical mechanism
that would influence each contiguous cell – a very elaborate piece of machinery. He was
trying to identify the electric and magnetic fields.
The electric field (E) depends on position and time – same thing for B. His device had
some kind of representation of E&B fields. Then he wrote down Newton’s laws of
motion to see how the E&B fields behaved as a function of time. He was able to derive
equations that represented the first and second laws that we wrote down. The third law
was also easily accommodated.
In the fourth law, he noticed something strange. He always found an additional term. The
experience was that there was this additional term, an amendment to this fourth law. It
contained a term proportional to the rate of change of the electric charge. In other words,
the field B was generated not only by an electric current, but by an additional effect in
this fourth equation that he derived. Proportional to the rate of change of the electric
field. He was persuaded that this was an error, at first.
He observed that the fields would have to be very rapidly changing before that term
would give any observable effects. There could indeed be such a law present, and it never
would have been directly observed up until that time.
Taking it seriously, he assumed that this amended law was the “true law,” and looked
carefully at the equations. As far as all the EB experiments of the day, this didn’t change
anything. Suppose we take this additional term seriously, it has a certain elegance. It
restores a kind of mathematical symmetry between the E&B fields.
(29:45) The first two laws are really the same law (inverse square law for forces), the
difference is that in the second law there are no isolated magnetic poles. And it restored
symmetry between the 3rd and 4th laws.
It turns out that we need a statement of this sort to uphold the law of the conservation of
electric charge. In fact it would have been possible to raise considerable objection to the
laws prior to Maxwell because they are not consistent with the laws of the conservation
of electric charge. Maxwell’s term restored the law of conservation of charge.
The strange and wonderful thing is this: the equations were changed in their
mathematical character by this term that Maxwell added. They began to possess solutions
that they never would have had before. Previously, if you took the equations as they
stood earlier and put down electric charges, you would get fields that dropped off very
rapidly with distance.
If you take an electric dipole, let them oscillate in position (oscillating dipole), according
to the first set of equations (without Maxwell) would produce an electric field that would
fall of as 1/r cubed.
What Maxwell found was that there are additional effects which are introduced by his
amendment term that lead to the possibility that you can have E&B fields in free space,
even without the presence of sources.
When he looked at the dipole situation, he found that the amplitude of the oscillating
fields only went down as 1/r as opposed to 1/r cubed. We’re talking about the radiation of
energy into free space.
The simplest form of the equations that Maxwell could solve where the ones where
charges and currents were set to zero. Was it possible to have E&B fields simply in free
space? The answer he found was yes, as long as they are oscillating, and as long as they
have orientations relative to one another which are the ones prescribed by his equations.
He found that these solutions to the Maxwell equations were in the forms of oscillating
waves. He could write down the equation that shows you how these waves propagate, and
constants that had already been measured regarding the Coulomb law and the like, they
told him what the velocity of his waves were. He found that the velocity of those waves
was 3.10 x 108 m/s. And that agreed with the precisely known and measured velocity of
light. In other words the equations predicted the existence of waves traveling through free
space containing oscillating electric and magnetic fields, and that’s even in the absence of
electric charges and currents.
The simplest solution that he found was the plane wave. The representation of these
waves is given on the slides:
(42:00) There was no experimental confirmation for Maxwell’s predictions. He assumed
that he had explained the nature of light, but there was no experimental confirmation of
his amendment to the fourth law.
That changed in 1887, with the famous discovery of Heinrich Hertz. What happened was
this: he had a device generating sparks in his lab. This device happened to be connected
to a loop of wire. He found that each spark has a rapidly oscillating current that generated
the EB waves radiated by the coil.
The second loop is the antenna detecting the radio waves emitted by the first. As soon as
he gets any distance away we get no more reception. This is where the change in our lives
really began. The simplest EB waves are these plane waves that we saw portrayed on the
screen. One of the things that Maxwell found that there is energy being put into the EB
field. This was consistent with his notion that the vacuum was an elastic medium.
Our lives have been radically altered by radio waves.
Maxwell found that there is an energy density communicated somehow through the
vacuum. Actual energy is being transported in space, and it is being transported in fact by
the velocity of light. There is also a momentum density (U/c).
Decades later, to everyone’s surprise, it was discovered that there is a particle nature to
waves.
But we are still dealing with the picture of light as continuous waves.
(49:00) Demonstration: Microwaves
All of these waves are essentially the same in structure, even though there is an enormous
amount of frequencies available. But they are fundamentally are the same.
He holds a bulb in front of the microwave generator, and the bulb lights out. The
radiation that comes out is already polarized. Polarization is nothing but axis along which
energy radiates. The bulb is sensitive to the electric and not magnetic field. There is just
as much electrical energy in a plane wave as there is magnetic, but devices don’t usually
respond to the magnetic field.
4/21/06
We’ve talked some about the remarkable discovery made by Maxwell: the important
thing to emphasize is that it was a purely theoretical development. It was incredible that
by a thought process (however bizarre) one should be able to construct a new law of
nature, and then have the discovery made that in fact one had invented a correct natural
law. He was empirically proven correct twenty years after first formulating his law.
This law had a number of important consequences, some even before electro/magnetic
waves were discovered as such in the laboratory. The first of them was Maxwell’s
observation that these waves represent an altogether new solution to the equations
describing the basic laws. According to the pre-Maxwell laws of electricity and
magnetism, charges and moving charges representing currents, would indeed have
electric and magnetic fields surrounding them, which would typically drop off rapidly
with distance. There was something rather localized about those electric and magnetic
fields, even though they did reach out for some distance: if you moved the charges or the
current-bearing wires, those fields would move with them. Fine, but they were still
localized (in much the same way).
The terms that Maxwell added to the electromagnetic equations changed the character of
the solution to those equations in a very fundamental way. They made it possible for
fields to become detached from their sources and to move freely. Maxwell observed that
they would have to be oscillating fields to move freely in space. The speed with which
those waves would move, turned out on the basis of existing data, would have to be the
measured speed of light. So Maxwell was immediately convinced that he had discovered
the electromagnetic basis of light, that light consists of electromagnetic waves, and he
became persuaded that electromagnetic waves could also be generated in the lab with
larger scale equipment. It was just a question of securing sufficiently rapidly oscillating
currents/charges in whatever electric circuits one was using.
That wasn’t easy to do, and in fact it happened in a somewhat accidental way in Hertz’s
laboratory. Sparks involve rapidly oscillating fields, there was a rapidly oscillating field
implicit in each spark.
(10:00) Maxwell looked further at the way in which these fields become detached from
their sources. Even with his revised set of equations, a uniformly moving electric charge
will of course take an electric field with it, but it will not radiate any energy (as long as it
is moving uniformly). If you move a charge uniformly the entire field with just move
uniformly with the charge. That does not constitute radiation.
In order to have energy radiated into space, and to have it go off with the speed of light,
you have to accelerate charges. It’s only the acceleration that will do it. Uniform motion
doesn’t do it. When you accelerate charges, in fact, a certain amount of energy gets
radiated away and therefore costs a little extra energy to accelerate the charges.
Last time: the electromagnetic spear. What this is intended to show is a skeleton of a 3-D
plane wave. You have to imagine the wave coming toward you so the propagation vector
is the black arrow pointing toward you. The simplest plane waves that Maxwell could
describe were ones in which the electric and magnetic fields are oscillating necessarily
because static fields don’t radiate. These are fields which if you observed at any one point
are oscillating as a function of time, and on the other hand, if you could take a snapshot
of the wave, its skeleton would look like the spear he is holding. The red pegs indicate an
electric field that vary with position (pos/neg); and there is also a magnetic field that is
perpendicular to the direction of propagation (it is perpendicular to the electric field as
well). Now you have to imagine that this thing fills out all of three dimensions, and
whatever you see here is taking place over planes perpendicular to the axis. The same
thing is going on everywhere in a given plane perpendicular to the direction of
propagation.
(17:00) Around the time of the Second World War there was a radar generator that was
devised to oscillate at a frequency appropriate to waves of about 10cm wavelength. Last
time we observed that the “magnetron” does radiate.
The field is radiated from the horn. The horn makes a noise. There is a detector, a tiny
antenna, that detects the wave coming out of the horn and the antenna feeds into a little
device which sends the signal on to an amplifier. We demonstrate that this is a polarized
wave that emerges.
What do we mean by polarization? By a kind of common accord, we can take the
direction of the electric field to represent the polarization of a transverse wave. Why is
that? Because the electric field is the thing that primarily pushes charges around. The
magnetic field doesn’t do very much to electric charges. If the charges are standing still it
won’t do anything. Detectors in general respond to the electric field. And it is the
transverse electric field that represents the direction of polarization. Light is a transverse
wave.
The setup we have in this demonstration is a physical analogue of a sheet of Polaroid.
The Polaroid material was responsible for polarizing light. We don’t have to polarize the
light coming out of the horn; it’s already polarized. When he rotates the analogue of a
Polaroid filter, there are instances of very little radiation. We can see that the electric
field is oscillating in a horizontal direction. As he rotates the disc, it shuts off the field,
which tells us that the electric field is oscillating in a horizontal direction.
(22:00) Reflection from a metal plane: Dan holds a metal sheet in front of the magnetron,
after the professor removes the analogous “Polaroid filter.” He holds what looks like a
stick in front of the machine, and it glows when the wave hits it. We can clearly see the
reflection taking place as Dan rotates the sheet of metal.
He then sets up a standing wave. Remember that if the wavelength is 10cm, the distance
between the nodes in a standing wave will be 5 cm.
(25:00) Double Slit Interference Experiment
We’re going to pass the micro-wave beam through two slits cut into a metal sheet. We
see (hear) is that it is the single slit pattern. We can hear reinforcement as Dan rotates the
receiver along circumference of the circle.
(26:57) The Prism
Now, we are using the prism. Dan sets up a huge block of paraffin (transparent to
microwaves and it refracts them).
As observed that you need accelerated charges in order to create electromagnetic waves.
The first waves talked about were plane waves that can exist in free space and travel
indefinitely once they become free of their sources. He wanted to ask in greater detail
how this energy is sent into the vacuum. He found that there is an energy density in
space, present even in electro-static fields, but that gets carried away in radiated fields
Energy Density = ν ~ E2 + B2
The amount of energy per unit volume of space is proportional to the sum of squares.
There is just as much energy in a plane wave as there is electrical energy. But the
magnetic energy is not what pushes electric charges around (unless they are already in
motion). If they are stationary, you can forget about the effects of that magnetic field.
When we began discussing electric and magnetic fields they were hypothetical
constructions. We talked about the forces between the source of a field and a test charge.
We imagined putting that test charge at every single point in space in order to measure
the electric field, or putting a test pole at every point in space to measure the magnetic
field – these were completely theoretical constructions. In a sense, it didn’t make that
much difference as long as those objects were local to their sources. But that ceased to be
true with Maxwell’s equations. And you have to begin to think of EM fields as things. Of
course they are not material, but they are things that can take off and move away into
space.
The EM field in other words is a species of thing that we have to deal with in space.
Space is full of these fields (stars). This is a very real thing, even though it’s not matter.
(35:00) Maxwell talked at some length about the simplest form of radiation by an
accelerated charge that he could imagine. He takes a fixed (negative) charge at one
position, and a positive charge that oscillates back and forth. The name for this is
oscillating dipole. Another way to think about the same situation is to take two charges,
one plus and one minus, and have them vary harmonically in strength. Still another way
of securing an oscillating dipole is to let the charges be fixed in magnitude and let them
both just oscillate back and forth. In a mathematical sense, these are all oscillating dipoles
and they will all radiate in the same way.
(36:00) What Maxwell found was that there will be radiation that goes off in all
directions except one – that direction is along the axis.
Explanation of the diagram: we see oscillations taking place along the x-axes. There will
be radiation that goes off in all directions except one. That direction is along the axis.
From a kind of symmetry that there wouldn’t be any way of emitting radiation along the
x-axis because the radiation that goes off has to have its EM fields perpendicular to the
progress of the wave and perpendicular to each other.
No radiation whatsoever is radiated along the axis of oscillation. What you find is that the
amplitude of the waves that emerge depends on the direction/angle at which you are
looking relative to that axis of oscillation.
(39:00) What’s found is that the E field and the M field both have an amplitude
proportional to the sine of theta divided by the distance. It’s a 1/r falloff of these
oscillating fields. They of course have a time dependence as well, to indicate that they are
oscillating with time as well.
The characteristic radiation pattern then of dipole radiation – you have to square these
fields. The energy is conserved as you go out to larger and larger radii. The surface of a
sphere increases as r squared. So the amount of energy on a large sphere is just as great as
the amount of energy near the surface of a smaller sphere.
What this means is that no energy is lost when this radiation takes place. You have
just as much energy on the average going through spheres of any radius however
large.
This dipole pattern is something that we see many instances of: an antenna is a piece of
wire that is connected to a generator to have an oscillating charge (dipole). This radiates
in all directions. Its maximal intensity is radiated in the horizontal plane, while no
radiation in the vertical plane.
(43:00) He uses a slinky to make a crude antenna.
When do we see dipole radiations? The most primitive radio antennas were linear in
structure. They have become much more complicated since then. For example, if you
wanted your broadcast to be heard maximally, the right thing to do would be to have a
vertical antenna because then the wave would spread around the area perpendicular to
that direction.
See written notes.
Also, almost all scattered light is dipole radiation. Let’s suppose that we have a plane
wave in which the electric field is radiating in a given direction. It strikes a small,
electrically neutral object (like a dust particle). What does that electric field do? Well it
impels a little motion of the charges in this electrically neutral particle and if it’s going to
impel the motion, then at one instant of time then the particle is charged pos/neg on one
side, and then it reverses. A certain amount of motion is induced in the charges in a
neutral object. That turns the dust particle into an oscillating dipole, which will then
radiate in just the way Maxwell described – a lot of light in the horizontal plane (zero
light in the vertical direction). This light that is re-radiated, is scattered light.
Why is the sky blue? Because the air molecules are doing precisely this – dipole radiation
and re-radiation by individual molecules.
We turn to the demonstrations for this using Cambridge tap water. We have an aquarium
filled with tap water. We see a beam of light that is projected into the tank. There is also a
mirror to the back of the setup. The beam is simply re-radiated dipole oscillation.
We are going to polarize the light and observe its scattering. He adds a couple of drops of
milk. He then mixes up the water a bit. You can see the beam from the two angles a lot
more clearly. He puts a polarizing filter on the light beam by the tank. If he polarizes the
light one of those two scattered beams (mirror and tank), as one grows bright the other
grows faint. As he rotates the polarizer by 90 degrees the two switch positions.
We are seeing simply the effect of the dipole radiation pattern. We send in vertical
polarization, that makes the charges oscillate up and down, they re-radiate maximally in
the horizontal plane, and you see strong scattering. Then he rotates the polarizer so that
the light sent in is oscillating in a horizontal plane, and so the charges in the milk
particles are oscillating along the line of sight, and thus they can’t radiate in that
direction, so we can’t see them.
(53:00) Then he pours in a lot more milk and mixes it around. The blue light is being
scattered rather more than the red light, and you can see the effect of that in the reddening
of the light on the door. That’s simply what is left because a larger proportion of the blue
light has been scattered out of the beam. Multiple scattering is taking place, and we get
this dipole effect only when single scattering takes place.
Lecture 4/24/06
So far we’ve described the magnificent inspiration of Maxwell, which in one stroke
explained the nature of light and a great deal more. It wasn’t for another 20 years or so
that EM waves were realized in the lab by means of large-scale equipment.
The EM theory opened up a whole new field, and also meant a kind of turn in the
generations of scientists. There are a lot stories about old scientists that have trouble with
the theories of the youngsters. Maxwell’s theories meant that one suddenly had to know
more about partial differential equations that wasn’t terribly well-developed at the time.
That put quite a strain on the old men of his day. The new subject was taken up by
younger people, who began thinking about much more seriously about the microscopic
constitution of matter.
Not much was known of that in the 1880’s and 1890’s. But one thing was evident: even
when matter was electrically neutral (in the sense that there was just as much pos as neg
charge) it was not completely inactive/inert. We saw examples of that in the last lecture
by particles of milk suspended in the water tank.
Any bit of neutral matter, placed in an electric field, is distorted to some extent
electrically. You don’t actually see distortion. If we have little cube, for instance,
composed of matter, and we put it in an electric field, there will be a very tiny bit of
migration of charged particles within the matter, which are pulled out of place by the
electric field. (see written notes)
(10:00) The reason that he uses the term distortion is because technically the process is
dialectric polarization, but it’s also called distortion. This is not to be confused with the
polarization of EM waves - which is a property of the waves. In an EM wave, the
direction of polarization is the direction of the oscillating field that impels charges to
move. And so one has these two rather different uses of the word polarization.
In any tiny bit of matter, a certain amount of electrical polarization is induced by the
oscillating electric field in an EM wave. We have seen that this oscillating electric charge
is in effect what we call an oscillating dipole. It’s electrically neutral, but nonetheless
there is the production of an oscillating electrical field which is felt at great distances.
This little cube of matter becomes an oscillating dipole, which involves accelerated
charges. Where is the acceleration in this? These charges are oscillating harmonically up
and down in the cube. This accelerated motion gives rise to dipole radiation. The
radiation goes off, essentially in all directions. Suffice it to say that this little cube of
matter is re-radiating in essentially all directions; scattering.
What has this to do with the behavior with large-scale matter? Suppose we have glass
with an EM wave incident on it. What is going on in the glass?
Here is a picture of what’s going on in the glass (see notes). Think of any little cube of
glass (one tiny element of millions of such cubes). What is happening is that the electric
field of the wave is inducing scattering by the tiny cube. The tiny cube is re-radiating in
all directions, and you have to imagine that every one of these little cubes of glass is
doing the same thing. So you have all of these tiny elementary volumes of glass reradiating in all directions. This is very reminiscent of something we talked about earlier
in the term:
Huygens had a kind of model for the propagation of waves. He said that in a medium just
as on the surface of a pool of water, when you start an oscillatory motion, every single
point at which that oscillatory motion is taking place is a source of induced oscillatory
motion of all the neighboring points. This gave a way of constructing wave-fronts. In
effect what Huygens had was a vision of the actual explanation of refraction.
Refraction by glass, results from the superposition of all of the scatterings that are taking
place from those elementary cubes. In other words, if you have one particular cube, and it
is becoming an oscillating dipole, all the other cubes are doing the same thing. You add
up the waves to get the wave fronts. (not sure about this one).
So refraction is in effect a superposition of scatterings. And these scatterings are not
chaotic as they are in a cloud, or in the suspension of milk. All of the cubes of glass are
alongside one another. All of their contributions add up coherently to produce a very
simple result – the progress of the wave going through the medium as though there were
nothing there to disturb it.
Huygens simply assumed that the wave propagation velocity is different in glass from air
or in vacuum. He just assumed that, but it was verified and we know it’s true. But now
we’re in a position to ask why that’s so. The answer is because this dialectric polarization
that is produced, the tiny separation of pos and neg charge, has a bit of a time lag in it, a
kind of inertia.
When the electric field is impelling that motion of the charges, the motion of the charges
doesn’t precisely keep up with the electric field. It certainly has the same frequency, but
it lags by a fraction of a cycle (short time). As you have a time lag, between the response
of any of these volumes to the incident wave, that lag keeps accumulating. The
accumulation of those time lags is the slowing down of light, the reason that light goes
slower in refraction than in the vacuum.
What goes on when we have refraction – a very complicated picture of scattering, where
the scattered waves coherently add up when you go from one volume to the next.
(21:00) There is an interesting sense in which we can confirm that. Remember that dipole
radiation does not take place along the direction in which the charges are accelerated.
Instead it has this funny donut shaped pattern of the polar pattern. The radiation is the
strongest in the plane perpendicular to that direction. And there is no radiation
whatsoever that takes place along the axis of acceleration in the dipole.
Well now this ought to relate to our discussions of polarization of light waves. Remember
Brewster’s law. It had to do with polarization by reflection. When you had light polarized
in the plane that contains the normal. He takes up the “spear.” You can forget about the
magnetic part of the light wave, but it’s the electric field that accelerates the charges. In
this picture, that of Brewster’s angle, we are showing a light wave coming down on a
surface, with two possible polarizations. Polarization takes place in the plane
perpendicular to the surface. Now we have to agree that there is this time lag due to
inertia, due to velocity change.
So this wave is indeed going to be refracted, as shown. But now the interesting thing is
this: all of this refraction takes place because of re-radiation through the scattering. Every
elementary volume in the glass is scattering the light, re-radiating the light. We have to
imagine oscillating dipoles. But they will not radiate in the direction in which the electric
charge is oscillating. That corresponds precisely to Brewster’s determination that you
secure complete polarization of this radiation, that nothing gets re-radiated/reflected,
when the reflected wave would be perpendicular to the direction of the refracted wave.
(26:30)
It may sound complicated, but it’s an elegant confirmation of the theory we’ve been
talking about. You have Brewster’s law, by which you secure polarization by reflection,
confirming our theory that refraction consists of the added up effects of re-radiation.
Many things have been explained this way, almost all of the electrical behavior or matter
(some of it really complicated) and we don’t have the time nor the math to go into it in
great detail. By the 1890’s the understandings of the electrical behavior of matter were in
good shape. It was only then that things began changing materially, and quite a few
surprises popped up.
One of the passions of the 19th century was the study of electrical discharges in gases. We
saw instances of this in the hydrogen and neon tubes earlier this semester. Around the
middle of the 19th century the people that developed the techniques of evacuating glass
bulbs good really good at what they did. By the 1850’s it became possible to take the
pressure down to a ten-thousandth that of atmospheric pressure.
They often had electrical discharges going on in them. Those discharges showed the
colors characteristic of whatever colors were left. (30:00 – sound dies)
4/28/06 A29 Lecture
Sometime near the end of the last lecture, we were learning some things about the EB
spectrum It is indeed a remarkable kind of resource, one that was never known of before
the days of Maxwell. It’s a relatively recent discovery, but it is impt in our lives for the
communication of information.
There are no natural limits on the available EB spectrum. In practical terms, it takes an
enormous antenna to radiate long wavelength waves. We do have experience with waves
with wavelengths of 100 km. On the other side of the spectrum we have gamma rays. The
ones that we can create are created in particle accelerators. We also detect gamma rays
coming from out space.
There is quite an enormous range of frequencies in EB spectrum. There is an awful lot of
the EB spectrum that we do not see. We see less than 2 in a total of 76 “octaves,” and
who knows how many more octaves are possible at either end.
The remarkable thing is that they are all described by the Maxwell Equations. There is an
additional, very important, addition that we should make: at the higher frequency/shorter
wavelength end of the spectrum, what we see are individual quanta.
The spectrum can be divided into pieces according to the different technologies that you
have to take advantage of in order to handle electrical oscillations of those particular
frequencies.
There is this enormous range of magnitudes of frequencies and wavelengths, which
makes the various regions of the EB spectrum behave differently from one another.
Once you get to microwaves, you can’t really build electrical circuits that small very
easily, and so molecules do the infrared oscillations. Eventually the electrons in atoms do
the radiation as you get smaller. As you get smaller wavelengths, you go deeper into
the atom
Each part of the spectrum has a different “personality,” and thus behavior. Beyond
microwave frequencies, we can’t really control the radiation. We take the radiation as the
molecule gives them to us.
The laser “tames” the oscillations of atoms. It makes large quantities of atoms oscillate in
unity – hence coherence. But we don’t have UV and X-ray lasers.
We use these waves for communication: radio and microwaves have been used for quite a
long time now. The interesting thing is to the visible light bandwidth.
(20:00) So how do we send these EB signals? Radio broadcasting. All radio stations send
out a fundamental wave (the “carrier wave”). The CW tells you that someone is alive at
the radio station and it is on. Nowadays we send all kinds of info out on radio waves.
How do we do this?
We send out the carrier wave, but we modify it. In AM radio, a carrier wave has a
frequency of tens of thousands of Hz (cycles per second). These are frequencies that are a
great deal higher than the voice or musical frequency of the broadcast material. So what
is done is to modulate the amplitude of the carrier wave to make it increase and decrease
following the acoustic wave that you want to send. That acoustic wave is turned into an
electrical oscillation through a microphone. The electrical oscillation is used to amplify
or decrease the amplitude of the carrier wave (hence the name amplitude modulation or
AM radio).
This is/was the most primitive form of broadcasting, and it was based around the
modulation of the carrier wave. Your radio device would respond to the carrier wave. The
receiver had devices in it that made it pay attention only to the modulations (difference in
frequency). Maxwell found that you have EB waves when they were relatively high
frequency. If Maxwell had looked for EB waves at acoustic frequencies they never would
have found them, because EB waves require high frequency in order to transmit.
(25:00) Once you have transmitted the wave, no one cares about the rapid oscillations in
the carrier, attention is only paid to the modulations. At 25:00 or so there is the definition
of this modulation in mathematical terms: the basic oscillation of the carrier frequency
varies in amplitude with time. We have the product of two cosines because of the
modulation. When we modulate the carrier wave periodically we actually have three
waves present simultaneously (the carrier wave, and two others added by the modulation
(they are called “sidebands”)).
The cost of using this carrier wave to transmit the information in our acoustic wave is that
we excite the field not just at one frequency but at three. If we are playing music at a
radio station, for example, we want to convey a great many frequencies. A radio station
does not just take up a single point – you have to allot that radio station a band of
frequencies. The FCC was set up just for this purpose, allotting finite frequency band =
5000 cycles per second. That limits the possibility of sending hi fidelity sound over
amplitude modulation. They are legally required to throw away all those oscillations that
represent true hi-fidelity. FM radio is broadcasting at a much higher frequency, and so
there is in a sense much more frequency available, and they are not so worried about the
crowding of radio stations on the frequency band.
The frequency spectrum is a bit like valuable real estate: the people who have or have
been given the right to broadcast in a particular frequency channel. There is only one
“ether,” and one possibility within a given band of frequencies. This implies a monopoly
over that. There are analogies to pollution problems. The EB spectrum has a virtue
however, but it doesn’t last. You broadcast terrible stuff, but it goes off into the universe.
There is a good deal of scientific investigation into the TV and radio programs of ET’s
that we should be able to pick up as well. All in all, there is a solution to the problem. We
can crowd all the signals that we are sending out into the ether into optical fibers. The
information-bearing capacity of optical cables is comparable to that of the ether.
One further property he points out: unity. Waves at one end of the spectrum, can always
be converted into waves of the other. How? The Doppler Effect (changes the
frequencies of sounds). This effect exists in the EB spectrum as well, all though it is
usually extremely small and tough to measure.
We have gone as far as we’ll go with the “classical theory of light.” Although there have
since been a lot of developments in light theory, they are just elaborating on Maxwell’s
laws.
A subject of great interest to everyone is heat and thermodynamics. There was a certain
understanding that matter consists of molecules in a gas. Not that much was understood
about the structure of solids and liquids. One of the objects of study was heat radiation.
If he passes a current through a filament it glows red, and if he passes more current
through it the temp rises. At each temp. there is a continuous spectrum of radiation of
light. Spectra change as you alter the temperature. The intensity of the spectrum is a
function of frequency. As the temperature is raised more radiation is given out (much
more in fact), and the spectrum extends more and more into the visible.
There is something very basic about it. To the extent that heat has something to do with
the motion of molecules, there should be some hope for understanding the shape of the
spectral intensity pattern. One of the possibilities was this: for every mode of oscillation
of the EB field (normal modes of oscillation of sound, but there are normal modes of
oscillation for EB field) if one puts and equal amount of thermal energy into each normal
mode you can derive an expression the energy density at any frequency. The problem is
that as you go to higher frequencies there are more and more modes available for you to
put energy into (a law that was bound to go crazy).
There was also a guess: you have a drop off at the high frequencies. The interesting thing
about this distribution (black bodies). It was shown thermodynamically that there must be
something universal about the shape of these curves. At any temp. they have a unique
shape. Were it not so you could construct a perpetual motion machine.
In 1900 Max Planck had the notion that he could find the formula for the radiation
distribution. He had no basis whatsoever for that formula (it was interpolation). He had to
adjust one constant in that equation, h. Then it finally fit the data. He found the formula
almost magical in its fit to the data. He tried to find a derivation for that formula as an
expression of thermal equilibrium. He knew that the nature of the matter that was hot
didn’t matter much as long as it was able to exchange energy with the EB field. He knew
the interaction of the EB field with harmonic oscillators. He said let us consider matter
that consists of harmonic oscillators. He never could derive his magic formula as an
expression of thermal equilibrium unless he made a remarkable assumption.
What was the assumption? He was talking about harmonic oscillators interacting with EB
field. You can give to this oscillator any energy that you like. What he found was that
when you are dealing with the interaction of light and harmonic oscillators that the H.O’s
cannot have just any energy. Their energies are integer multiples of the quantum (the
constant h times the frequency). The energy of a harmonic oscillator has to be an integer
times this quantum. He said that the notion that energy is a continuous variable for this
oscillator is an illusion. It’s because we are dealing in exciting this oscillator with
astronomical numbers of quanta and we can’t determine one quanta more or less. At the
atomic level, the story is very different however.
How did that explain this radiation distribution?
If the frequency of the oscillator is very small these energy levels are going to be closely
spaced. If you have higher frequency oscillators, those energy levels are going to have
larger intervals of energy between them.
What is thermal excitation?
It is very limited. When a gas is raised to a temperature, the energies of the molecules
vary in a specific range. If you say that these harmonic oscillators are excited to a certain
temp. that will allow for the excitation of several/many of these energy levels.
But a high frequency oscillator, that thermal excitation will never even get us up to the
first step. High frequency oscillators never get excited and they never participate in the
radiation process.
Explains why it is that the radiation spectrum has the particular form that it does.
He was able to derive his magically effective formuala for the shape of the radiation
spectrum as a function of temperature. He never quite believed the result.
Next is Einstein.
Lecture 5/3/06
N. Bohr
L. de Broglie
C.J. Davisson & L.H. Germer, GP Thomson
W. Heisenberg
J.J. Baumer
E. Schrodinger
Compton – X-rays behave like particles
The date we’ve gotten to – 1911. Rutherford and his corps of graduate students produced
these observations about the deflection of alpha particles passing through thin gold foils.
This gave us the signature of Coulomb’s scattering. The alpha particles would not be
scattered very much by electrons.
The law of deflection ( the number of alpha particles suffering deflections) corresponded
precisely to scattering by a point positive charge. The alpha particles come in various
positions relative to the charge, and they suffer unique deflection along a hyperbolic path
depending on from where they approached the nucleus.
They knew the size of atoms to be around 1 angstrom, or ten to the minus ten meters.
Mendeleev – you could use the atomic number and make sequences of elements. Used
chemical behavior or atoms to observe the periodicity of atoms.
One of the more important elements of Rutherford’s discovery was that the atomic
number corresponded to the number of positive charges (charge opposite to the electron)
in the nucleus.
The question was where were the electrons? We got no information from that from
Rutherford’s experiments.
One of Rutherford’s grad students was a Dane named Niels Bohr. Bohr began thinking
about what the structure of atoms might be from an electronic standpoint. One of the first
things to occur was that these negatively charged electrons are attracted to the nucleus,
much like the planets are attracted to the sun. The electrons have a very strong Coulomb
force between them.
The difficult part was: if these electrons are running around in orbits of any sort (circular)
they are accelerated – uniform circular motion always involves acceleration. Accelerated
charges radiate. If the electrons are going around nuclei then they must be radiating
(losing energy) and getting closer to the nucleus. The atom would then collapse and the
electrons would give off x-rays.
Bohr realized that there must be something wrong with that picture. There is a good
reason why atomic energy states are stable and don’t collapse in a burst of X-Rays. It was
also clear that atoms must exist in states of discrete energy.
Planck assumed that harmonic oscillators can exist in uniformly spaced energy states.
Bohr said that something is stabilizing a sequence of energies for atoms. He didn’t know
what it was, but he tried to characterize it.
(20:00) If you have two different possible energy states for an atom, and that the atom
can go from high to low energy states. Bohr frequency conservation: as an atom goes
from a higher to a lower state (or visa versa) then it gives off a photon (energy: hv). It
contains the observation that there are these stationary energy states of atoms.
Bohr was looking for stable orbits.
Demonstration: he takes a ball and swings it around a pole. As the string gets smaller the
ball speeds up (conservation of angular momentum). Think tether ball.
This gave us all of the spectrum lines for hydrogen. When you deal with Helium and
Lithium. It was a spectacular theory for understanding atoms with a single electron.
De Broglie – made a remarkable suggestion: matter behaves like waves. De Broglie
asked how would it be if we associated with massive particles a wavelength? He gave
wavelengths to massive particles. We know what the momentum of massive particles
is/are.
(51:00) Demonstration: There is a plastic ring glowing green, and then Dan sets it
vibrating. We see more nodes as the frequency increases. It’s a resonant phenomenon.
While it shows us what Debroglie was talking about, it’s not perfectly accurate for
showing what atoms look like. “Only certain wavelengths will fit around a circle.”
At first this was all totally crazy. Davisson and Germer, and Thomson were the ones that
first took up observing debroglie’s suggestions.
Can we ever see these waves associated with massive particles.