Download Notes follow and parts taken from Physics (6th Edition, Cutnell

Notes follow and parts taken from Physics (6th Edition, Cutnell & Johnson)
Electromagnetic Waves
Since we’ve seen so many connections between electricity and magnetism, it’s probably not
surprising to learn that physicists generally refer to them as part of the same force, known as the
electromagnetic force. A changing electric field will give rise to a changing magnetic field
(which causes a changing electric field, etc.) and we can observe some interesting effects when
we change the electric field in a certain way. To do this, we can just hook an AC generator up to
a pair of wires which will function as an antenna. As the generator’s emf builds up, positive
charges will build up on one of the antenna wires and negative charges will build up on the other
one. The emf will then decline until the wires have no net charge on them; after that, the sign of
the generator’s emf will change and the formerly-positive wire will start to accumulate a negative
charge, and vice versa. The electric field will point from positive to negative charges, so it will
be changing magnitude constantly, and changing sign (direction) twice per cycle.
As the field changes, what will happen to other charges which aren’t part of the antenna? They
will see an electric field that changes magnitude and direction regularly, but they won’t see it
happen instantly. The field spreads out from the antenna just like sound waves from an
explosion. The electric field always points in a direction perpendicular to the progress of the
wave carrying information about the field. There is also a magnetic component to the wave,
since the changing electric field will cause a changing magnetic field. The magnetic field will
oscillate at the same frequency as the electric field, and the two are in phase (i.e., they reach their
maxima at the same instant). The magnetic field is always perpendicular to both the electric field
and the direction of propagation of the wave. For this reason, the wave is called transverse – the
oscillations are perpendicular to the direction of travel. A transverse wave is what you would see
if you stretched a rope horizontally & tied it to something (or handed one end to someone like
you were both going to swing a jump rope) and then moved your hand up & down rapidly. The
rope itself moves up and down, but the wave will travel from you to the other end. If you’re
washing a car and the hose gets stuck under a tire, or you’re cleaning house and the vacuum
cleaner cord has gotten hung on something, you’ll probably try to free it by sending a transverse
wave down the hose/wire. Sending a longitudinal wave down the wire wouldn’t do you any
good. You need to be able to move it perpendicularly to its length, and that requires a transverse
wave.
Electromagnetic waves are different than just about any other kind of wave in that they do not
require a medium to “wave” in. Water waves need water, sound waves need air (or some kind of
matter), but electromagnetic waves can travel in a complete vacuum. In fact, they travel fastest
in a vacuum, and they all travel the same speed in a vacuum – 299,792,458 m/s or, more
commonly, 3 x 108 m/s. When we look at relativity, we’ll see that this speed has a special
significance.
When waves leave an antenna and travel through space, they can eventually strike another
antenna. The changing electric field can accelerate electrons in the receiving antenna and cause
them to move back and forth in the same way that the electrons in the sending antenna moved.
By reproducing this motion, we can reproduce the current that existed in the circuit attached to
the sending antenna. Of course, the reason we care about this is that the current in that sending
circuit will change in a very particular way with time. If it was independent of time, we
wouldn’t need an antenna – we’d just buy a battery and the right resistor, and we’d have our
constant current. We’ve seen that an LC circuit has a particular resonance frequency where it’s
easy to build up a large signal. If we have a fixed inductor and an adjustable capacitor, we can
“tune” that resonant frequency until we can pick up the signal broadcast by our favorite radio or
TV station.
The Electromagnetic Spectrum
The different colors of visible light that we can see represent different wavelengths (and
frequencies) of the changes in the magnetic and electric fields that make up light. The formula
connecting wavelength, frequency, and wave speed is the same as it was for sound, except that
the wave speed is now the speed of light instead of the speed of sound. That formula is
f  c
where f is the frequency of the light,  is its wavelength, and c represents the wave’s speed (we
generally use a c for light instead of v). For visible light, the wavelengths range from about 380
nm or so (violet) to about 750 nm (red). That means the light’s frequency ranges from 4 x 1014
Hz (red) to about 7.9 x 1014 Hz (violet).
The visible part of the spectrum is only a tiny slice of what’s out there. Moving from visible
light to lower frequency electromagnetic radiation, we get to infrared radiation first. This is
sensed (if it’s intense enough) as heat, and it’s what warms your hand if you put it near a light
bulb. It’s also what allows police departments & soldiers to see people in total darkness (night
vision goggles), and it’s what your remote control puts out. How far does this part of the
spectrum reach? There are no hard physical boundaries between any of the regions of the
spectrum – some are more definite than others because of biology (we can see red, we can’t see
infrared) and some by regulations (the FCC or some international standards body determines that
FM radio runs from 88 MHz to 108 MHz). As far as the physics is concerned, it’s a smooth
transition from low frequency to high frequency.
As we go to frequencies in the low end of the infrared, we eventually get to the microwave region
(around 109 – 1011 Hz or so). The microwaves that cook your food are in this range, as are most
cordless phones, cell phones, and wireless computer networking equipment. Microwaves are
sometimes lumped into what’s called the “radio region” of the spectrum. Also part of that is the
FM range mentioned above. Broadcast TV (which covers a much larger portion of the spectrum
than the 20 MHz or so allotted to FM radio) starts in the FM region and reaches up into the
microwave region (you can see this for yourself if you’re ever driving through an area that has a
local TV station broadcasting as channel 6 (not necessarily cable channel 6) – you’ll be able to
pick up the audio on the low end of your FM radio).
Even lower in frequency, we get to AM radio, which runs from about 550 kHz to about 1650
KHz or so. Below this, there are specialized transmitters which can penetrate deeper into the
ocean to communicate with submarines without requiring them to surface. Also at the low end
of the spectrum is the 60 Hz radiation produced by power lines (and just about everything
plugged into them).
What’s above the violet end of the visible spectrum? Immediately above it is the ultraviolet
range. This is responsible for giving people tans (and skin cancer), and reaches up to about 1017
Hz. Above that, we find X-rays like the kind used to look at your teeth and bones. These run up
to about 1020 Hz. Beyond that, the rest of the way up, we just talk about gamma rays. These are
some of the most potentially dangerous products from nuclear waste.
You might have noticed that it seems that the least damaging kinds of radiation are clustered at
one end of the spectrum (radio) and the most damaging (UV, X-ray, etc.) are at the other end.
That’s not a coincidence – we’ll see later that the energy of an individual piece of light (photon)
is proportional to its frequency. High frequencies = high energy photons which can damage us.
In a vacuum, all of these disturbances (collectively called EM radiation or, sometimes, just light)
travel with the same speed, which is 299,792,458 m/s. This speed can be measured with modern
equipment in an intermediate-level physics lab. Also, it can be derived from the laws which
govern electricity and magnetism, and it can be shown that the speed is connected to the
permeability and permittivity of free space by
1
c
 0 0
Energy of Electromagnetic Waves
In an LC circuit, energy sloshes back and forth between the magnetic field in the inductor and the
electric field of the capacitor. In fact, neither of these two devices is necessary to have energy in
the fields. Electromagnetic waves, which are composed of these alternating electric and
magnetic fields, also contain energy. A reasonable way to discuss this energy is through the use
of the energy density mentioned earlier. Using the same formulae, we can see that the electric
and magnetic energy densities will be

uE  0 E2
2
B2
uB 
2 0
uTotal  u E  u B
The energy in a wave is evenly distributed between electric and magnetic forms in these waves,
so uE = uB and we get
0 2 B 2
E 
2
2 0
E 2  c2 B 2
so E  cB
The peak field values (since they oscillate just like the voltage and current in an AC circuit) are
related to the rms field values as we would expect:
Erms 
E0
2
Brms 
B0
2
We will also be interested in the intensity of the waves, defined as the power per unit area
(watts/m2). Using our old relationship between energy and power, we define the intensity S as
S
P Energy

A
tA
For electromagnetic waves (which travel at c), the volume filled in time t will be ctA, so the
energy can be found by multiplying this volume by the energy density u, giving
uct A
c B2
2
S
 cu  0 E c 
tA
0
The Doppler Effect
If a source of electromagnetic radiation is moving with respect to a receiver of that radiation, they
will disagree on the frequency of the waves involved. This is a simpler version of the Doppler
effect for sound, which varies depending on who’s moving – source or observer. The reason it’s
different is that sound travels in air, so we can definitively say who’s moving (the person feeling
wind). Electromagnetic radiation doesn’t require a medium to travel in, and one of the basic
principles of relativity is that non-accelerating people will agree on the velocity of light, even if
they’re moving with respect to one another. In other words, you can’t measure your velocity
relative to the speed of light – it seems very weird, but you’ll always get c if you’re measuring it
in a vacuum, no matter how fast you are traveling with reference to something else (Earth, Sun,
spaceship, etc.).
The effect you will notice is a change in the frequency of the radiation as you encounter more (if
approaching) or fewer (if retreating) oscillations of the electric & magnetic fields. For velocities
which are small relative to the velocity of light, the change in frequency will be
 v 
f o  f s 1  rel 
c 

where the frequency measured by the sender (who is at rest relative to the source) is fs, the
frequency measured by the observer is fo, and vrel is their relative velocity. We get larger values
for frequency when moving towards the source and smaller values when moving away. The
drawing below shows this.
The number of peaks and valleys of the fields that hit our eyes in a given time determines the
color we see. If we move towards the source, or it moves towards us, we will encounter more
peaks and valleys per second than we would if we were standing still. That’s why we say the
frequency (which measures the number of peaks/valleys per second) increases as we move
towards the source, or it moves towards us. Similarly, if we move away from the source, fewer
peaks and valleys strike us per second than if we were standing still. The frequency of the light
is lower. In the scene below, this is what the pattern of peaks (or it could be valleys – either one)
would look like if the source was moving quickly. The source started at the center and was
emitting peaks and valleys – let’s say the circles represent peaks. Each circle is centered on the
point where the source was when it was emitted. The source is moving, so the centers of the
circles move, too. The blue face will be hit by lots of peaks in a short time, meaning a higher
frequency. The red face will be hit by fewer peaks in the same time, so it looks like a lower
frequency. There is no motion towards or away from the yellow face, so it sees the “real”
frequency of the source. This is the frequency that someone riding with the source (inside the
ambulance) would hear (or see, or whatever). By the way, the source in this figure is moving
half as fast as the waves themselves (that’s ½ the speed of light for light, or Mach 0.5 for sound).
The shift to a lower frequency is called redshift and the shift to a higher frequency is called
blueshift. These are universally used, but there’s something to keep in mind: we call it redshift
because, if we are talking about light which is in the middle of the visible spectrum, shifting to a
lower frequency means moving towards the red end of the spectrum. Similarly, moving to a
higher frequency means moving towards the blue end. The thing to keep in mind is this:
blueshift always means a shift to a higher frequency, and redshift always means a shift to a
lower frequency. As an example, if a source of radio waves is moving away from you, you still
say that it’s redshifted, even though it’s moving further away from the visible-light color red in
the spectrum.
Polarization
Electromagnetic waves only vibrate perpendicular to their direction of travel; that’s what makes
them transverse waves. Also, the electric and magnetic parts have to be perpendicular to one
another. This still leaves a great deal of freedom in the orientation of the fields, though. If we
imagine a wave traveling into the page, the electric field vector can be thought of as one of the
hands of a clock – it can be oriented in any direction around the full 360˚ of a circle on the page
(really 180˚, since a wave vibrating up and down is the same as one vibrating down and up). Of
course, if we specify a particular direction for the electric field, we’ve also specified the direction
for the magnetic field. The direction of the electric field determines the polarization of the
wave. Waves leaving an antenna will have a polarization which matches the antenna’s
orientation, but for high-frequency waves, the antenna is an atom. The atoms are constantly
being jostled by collisions, though, so the polarization of each emitted wave is different and the
net effect is unpolarized radiation.
Certain materials are used to create polarized light (or radiation) from unpolarized light. For
microwave radiation, a metal “picket fence” with pickets and spaces between them which are
smaller than one wavelength produce polarized radiation. For the optical range, the wavelengths
are so small that long chains of molecules serve as the fence. It would be natural to assume that
this arrangement blocks a great deal of light since only a small fraction of the light waves would
have electric field vectors which line up exactly, or almost so, with the spaces in the fence. One
of the oddities of quantum mechanics, though, says that there is essentially a coin-flip for each
encounter of a wave with the polarizer: the light must choose whether it is going to align parallel
with or perpendicular to the holes in the fence. The choice is random and half of the light gets
through, and half gets stopped.
Now that the light has made its choice, and only the light polarized in one direction is passed,
what will happen if we put another polarizer in its path?
It depends on the orientation of the second polarizer. If it is aligned in the same direction as the
first, all of the light going through the first should also go through the second (the light has
already made a choice about its direction, and all the light that chose to be perpendicular to the
first polarizer was stopped by it). What if the second polarizer is rotated by 90˚ with respect to
the first? No light will get through, because as we just realized, all of the light with electric
fields oscillating in the direction of the second polarizer got stopped at the first one.
What if the angle between the polarizers is not 90˚? Some light will get through, and some will
be stopped. Let’s choose the y-axis to be the direction of the first polarizer. The component of
the electric field in the direction of the second polarizer will be the electric field multiplied by the
cosine of the angle between the polarizers. Intensity, though, depends on the square of the
electric field, so the intensity will be
S  S 0 cos 2 
where S0 is the intensity of the light leaving the first polarizer (called the polarizer), S is the
intensity of the light leaving the second polarizer (called the analyzer), and  is the angle between
the two. This relation is known as Malus’ law, and it shows that an analyzer oriented parallel to
the polarizer shouldn’t change the intensity of transmitted light, but a 90˚ angle between the two
should block all light.
One of the very weird things about polarization is the fact that the “choice” a light wave makes
about its polarization can change. We know that two crossed polarizers will block all light, but
what if there’s a third polarizer between them oriented at some angle (other than 0˚ or 90˚) to the
first? Now, we’ll see light get through again! If we have a perfect barrier made by two fences,
we don’t expect the addition of a third fence in between the other two to suddenly allow passage.
Imagine the middle polarizer is oriented at a 45˚ angle to the other two. When light leaves the
first polarizer and is definitely up-down, it has to make a choice when it hits the middle polarizer:
realign parallel to the middle polarizer, or perpendicular to it. For a 45˚ angle, half will go
through and half will get stopped. The light that gets through now has the direction of the middle
polarizer. When it hits the third (and final one), it must again decide whether to realign parallel
to or perpendicular to the third polarizer. Again (for a 45˚ angle, since cos2(45˚) = ½), half the
light gets through, and half is stopped. The net result from all three polarizers is that half of the
light at each step is stopped by a polarizer, and (1/2)3 = 1/8, so 1/8 of the light is passed.
In addition to linear polarization, light can also be circularly polarized. In this situation, the
electric field vector rotates as it the light travels, and the wave can be called right or left-hand
polarized.
Wave Fronts & Rays
How does light interact with matter, and what can we do with that interaction? To study this,
imagine a tiny light bulb acting as a strobe light and sending pulses out regularly. As each pulse
of light expands from the source, it will grow into a larger and larger sphere. The surface of this
sphere is called a wave front. Imagine drawing lines of latitude and longitude on the spherical
wave front, and then concentrate on a small “square” of the surface (for example, the region
bounded by 0˚ latitude, 90˚ longitude, 10˚ latitude, and 80˚ longitude). If the sphere is the size of
the Earth, for example, this square will be reasonably flat. If we cut the square out of a baseballsized globe, though, it will be more sharply curved. We see this same effect everyday – a
parking lot may be hundreds m2 in area and seem very flat, while even a small piece of an orange
peel (a few cm2) won’t lay flat on a table.
As our wave front expands, it will seem to get more flat for this very reason. A truly flat wave
front would be called a plane wave, for obvious reasons. Lines drawn from the source to the
corners of our wave front are called rays. After our wave front has traveled a long distance and
is approximately a plane wave, the rays bounding it will be approximately parallel. Light from
the Sun has traveled so far that it is essentially parallel when it reaches Earth. Also, if you have a
laser pointer, the light coming out of it can be considered parallel. This is very different from the
situation with a flashlight – the spot of light gets larger (and dimmer) as you shine it on more and
more distant objects. The last thing we need to add to this idea is that it works even if we don’t
have a strobe light. We can consider the peaks of the light waves to be the individual wave
fronts.
Reflection of Light
If we have a flat, shiny object like an ordinary mirror, light bouncing off of it behaves in a very
simple way. If we measure the angle of the incoming light relative to the normal of the mirror (a
line perpendicular to the mirror), we’ll see that the angle of the incoming (or incident) beam is
the same as the angle of the exiting (or reflected) beam. If you think about this for a minute,
light striking a mirror behaves the same way as a basketball striking a hard wall, and for the same
reason.
Basketball has x and y components
of momentum – after striking wall,
y component is reversed but x
component is unaffected
Light strikes a mirror & the component
of motion perpendicular to the mirror is
reversed, but the component parallel to
the mirror is unchanged
y
x
i
r
This very simple law is called the law of reflection, and we can state it simply as
i   r
The way a mirror reflects light is known as specular reflection. The surface is very flat
everywhere, so neighboring rays are not scattered in random directions when they strike the
surface. For things which aren’t shiny, reflection is still happening, but it’s called diffuse
reflection. The surface is uneven on the small scale (remember that light waves are very small)
and the reflected light will still obey the law of reflection, but each different spot on the surface
has a normal pointing in a different direction. Imagine gathering many people into a gym
forming them into two lines. Give each pair of people a basketball, and have them bounce-pass
it to one another. It would be very easy to synchronize this so that everyone was doing the same
thing at the same time. This is like specular reflection. Now take these two lines of people out
to a bumpy patch of grass and try it. The basketballs will still obey the same physical laws, but
the uneven nature of the ground will send them in all directions (diffuse reflection).
Image Formation by a Plane Mirror
The ordinary mirrors we use every day are called plane mirrors because they are flat. Since
they’re flat, there is no magnification by a mirror like this. The image in a plane mirror appears
to be as far behind the mirror as you are in front of the mirror. Finally, while a mirror seems to
reverse right and left, it does not reverse up and down (see
http://math.ucr.edu/home/baez/physics/General/mirrors.html for an explanation of this effect).
The reason your image in a plane mirror is as distant from the mirror as you are boils down to the
fact that your eye always assumes light traveled in a straight line to get to it from an object. As
shown in the picture below, we can make a triangle out of the incident light ray, the mirror, and a
line perpendicular to the mirror running to the object. It will be geometrically similar to a
triangle made by the back of the mirror, a line perpendicular to it running to the apparent location
of the image, and the apparent direction of light coming from the image.
r
i

This is why it seems that the image in the mirror is mimicking your movements, getting closer
when you do and moving away when you do. This image is called a virtual image since it
appears to exist on the side of the mirror where no light is actually present (inside your medicine
cabinet, for example). Virtual images can’t be projected onto a screen, but the kind of mirrors
we’re about to investigate can produce real images.
Spherical Mirrors
So far, we’ve taken a look at what happens when light reflects from a perfectly flat, smooth
surface (a mirror), and a lumpy, irregular surface (a sheet of paper). What if the surface is
smooth, but not flat? One of the simplest kinds of curved mirrors is a spherical mirror. If we
cut a small section out of a mirrored sphere, we’ll get a spherical mirror. It can be shiny on the
outside, which will give us a convex mirror, or shiny on the inside, giving us a concave mirror.
Either way, we’ll talk about the mirror’s radius of curvature, which just tells us the radius of
the sphere it was cut from.
When light hits the spherical mirror, its angle of reflection will still match its angle of incidence
– the added interesting point is that the direction of the normal changes as we move around on
the mirror’s surface (just as it does on the Earth – people in Australia don’t see the same stars
overhead that we do because our definitions of “up” don’t match). If parallel light (solid arrows)
strikes a convex mirror, the reflected light (dotted arrows) will spread out in many directions (or
diverge) from a line (the principal axis) drawn from the center of the mirror to its center of
curvature (the location of the center of the sphere it was cut from) as shown below:
Principal
Axis
R
If the mirror is concave, light striking it will be directed towards the principal axis (that is, it will
converge) rather than away from it.
Principal
Axis
R
For light rays which are close to the principal axis (known as paraxial rays) and parallel to it
(paraxial rays aren’t necessarily parallel to the principal axis), all will focus at a particular point
on the principal axis. That point can be shown to be halfway between the mirror and its center of
curvature and is called the focal point. The distance to this point from the center of the mirror is
known as the focal length of the mirror, and we can write
1
f  R
2
where f is the focal length and R is the radius of curvature for a concave mirror. Rays which
strike farther from the center of the mirror don’t get focused to the same place (focal point) as
those hitting nearer the center. When these mirrors are used to form images, this effect (known
as spherical aberration) tends to blur the image. If the mirror is shaped like a parabola instead
of a sphere, this problem can be fixed, since all parallel rays which hit any point on the parabolic
mirror are focused to the same place.
Almost the same math works for a convex mirror, even though that light doesn’t get focused to a
point. Instead, the reflected light appears to diverge from a point behind the mirror. How far
behind? As above, only we write
f 
1
R
2
since the image is on the “dark” side of the mirror.
Image Formation by Spherical Mirrors
When light is focused by a spherical mirror, we would like to be able to find out things like how
far away the image forms, if it’s real or virtual, upside down or right-side up, magnified, etc. We
can do that by using the principles of ray tracing, which we’ll also use when we look at lenses.
This is an elegant, if not particularly fast or forgiving, way to locate images. We’ll see how to do
it mathematically later. There are 3 basic rules to follow, and if we measure our initial setup, we
can measure the intersection of the rays we trace and figure these things out. The three rules are:
1) Draw a line from the object (usually the top of the object) parallel to the principal axis that
intersects the mirror; from that point, draw a straight line which passes through the focal point
Principal
Axis
F
R
2) Draw a line from the object straight to (and through) the focal point on to the mirror; the
reflected ray is parallel to the principal axis
Principal
Axis
F
R
3) This line connects the object with the center of the imaginary sphere from which the mirror
was cut; when it strikes the mirror, it will double back and retrace its path
F
Principal
Axis
R
All three rays should converge at one point, which will be the matching point on the image (the
top of the image, in this case).
F
Principal
Axis
F
As you can see from the drawing, the image is upside down, smaller, and closer to the mirror
than the object itself was. The rays from the top of the object’s face all converge at the top of the
image’s face. Shortly, we’ll see how to predict these things without drawing a picture.
For convex mirrors, things are a little different since the center of curvature and the focal point
are on the opposite side of the mirror from the light rays. In this case, our rules are:
1) The first ray moves from the object (the top, in this case) parallel to the principal axis and
reflects from the mirror. After reflection, the ray will appear to have come from the focal point of
the mirror.
F
R
2) This ray is aimed from the object towards the focal point, and its reflection is parallel to the
principal axis.
F
R
3) The third ray goes towards the center of curvature and retraces its own path after reflection
F
R
We will again find the image at the intersection of these three rays, only this time we’ll find it
(apparently) behind the mirror, so it will be a virtual image.
F
R
Mirror Equation and Magnification Equation
By tracing the proper rays, we can see that the triangle made by an object, the principal axis of a
concave mirror, and a light ray connecting the top of the object to the mirror-principal axis
intersection will be similar to (that is, have the same angles as) the triangle made by the image,
principal axis, and light ray connecting the top of the image to the mirror- principal axis
intersection. In the image below, the green triangle made by the object is similar to the orange
triangle made by the image. If the triangles are similar, that means that the ratios of the sides are
equal. The sides are made up of the height of the object (or image) and its distance to the mirror
(we don’t care about the hypoteneuses right now)
F
Principal
Axis
If we call the object height ho, the image height –hi (the negative sign indicates it is upside
down), the object distance (always measured to the mirror) do and the image distance di, we can
write
ho d o

 hi d i
For a ray near the principal axis, we have another pair of similar triangles shown below (orange
and green, again)
Principal
Axis
F
The triangle’s lengths along the principal axis are (do – f) for the green triangle and just f for the
orange triangle. The heights are still ho, and –hi. Our second equation is
ho d o  f

 hi
f
which we can combine to get
1 1 1
 
do di f
which is known as the mirror equation (in the next chapter, it will also be known as the thin
lens equation).
We can use this same geometry to learn about the size of the image in relation to the size of the
object, or the magnification. Magnification numbers larger than 1 correspond to making the
image larger than the object, while numbers smaller than one represent a minification. The
formula is
m
hi
d
 i
ho
do
The rules we need to learn to use these equations are simple: First, all distances are measured
from the mirror itself; they’re positive for things in front of the mirror, known as real objects or
images, and negative for things behind the mirror (virtual objects/images). Next, focal length is
a positive number for a concave mirror and a negative number for a convex mirror. Finally,
negative magnifications represent images inverted with respect to the object, while positive
numbers mean the image and object are in the same orientation. That’s all there is to it.
Index of Refraction
While the speed of electromagnetic radiation in a vacuum is constant (3 x 108 m/s), the
interaction between light and atoms acts to slow the overall progress of the wave. Different
materials provide differing amounts of delay and therefore give differing speeds for the
movement of light. The measure of this “optical density” is called the index of refraction
(usually denoted by n). A vacuum has n = 1, and other things have n > 1. The connection
between c and v (real velocity through the medium) is simply:
v
c
c
or n 
n
v
Clearly, n has no units since it’s the ratio of two speeds. Some typical indices of refraction are:
diamond, 2.4; water, 1.33; glass (crown), 1.52; and air, 1.0003.
Snell’s Law and Refraction
The change in speed when light goes from one medium to another produces some interesting
effects. When light goes from air into water (low n to higher n), it will be bent at the interface
(unless it’s coming in along the normal to the interface). You can understand this by realizing
that one part of the wave slows down before the other, and that causes the turn. Think of the
light wave as a wheelchair going from a region of easy travel (concrete driveway) to a region
where it will move more slowly (grass). If you move from the driveway to the grass at an angle,
the wheel which touches the grass first will slow down, while the wheel still on concrete
maintains its speed. The net effect is to turn the wheelchair, as shown below (just the wheels and
axle are shown):
Wheel touching grass
slows down, other
wheel keeps moving,
so wheelchair turns
Notice that upon reaching the “slower” region, the wheelchair turns closer to the normal (a line
perpendicular to the interface between the grass and concrete – the normal would be up and
down the page in this case). Light behaves the same way. We’ll draw in some light rays and
include the normal because we want to be able to measure angles from the normal.
i
 rfl
 rfr
We have three rays in the picture; an incident ray, a reflected ray, and a refracted (or bent) ray.
This is the general situation when light hits the interface between two materials; some is reflected
and some is transmitted through it after being bent. We already know the relation between the
incident and reflected rays – they make equal angles with the normal, just as they did when we
looked at mirrors. The relationship between the refracted and incident rays is found in the form
of Snell’s Law, which we can state simply as
n1 Sin  1  n2 Sin  2
where n1 is the index of refraction in the first material, n2 is the index of refraction of the second
material, and 1 and 2 represent the angles (relative to the normal) made by the light rays in each
medium. Using this, if light strikes the surface of a swimming pool at an angle of 17˚ relative to
the normal, what angle will the transmitted (refracted) ray make with the normal? We just need
to solve Sin 17˚ = n2 Sin  for  (we’re taking n for air to be equal to 1 since it’s so close to one
anyway). We already know that the index of refraction for water is 1.33, so we get 12.7˚ for our
angle. Notice that we refer to materials 1 and 2, not to “incident” and “refracted” waves and
materials. That’s because we use the same math if you’re at the bottom of a pool with a
flashlight shining up into the air.
This effect is responsible for the strange appearance of a spoon in a glass of water – it looks bent,
even if it’s not. Why, then, can we look out windows and not see huge distortions? Not all of
the incoming light is perpendicular to the window panes, so it should be refracted and it should
distort our view. The neat thing is that the bending which happens going from air to the glass of
the window is reversed when the light goes back into the air from the glass. The light ray has
been displaced slightly, but it’s still going in the same direction it was.
You might have noticed that the inequality between n1 and n2 and the fact that the Sine function
can only range between –1 and 1. It seems we can write an equation that can’t be satisfied. For
example, what if we have a laser pointer under the surface of a swimming pool. If n1 = 1.33
(water) and n2 = 1 (vacuum, or air if we approximate) and our incident ray comes in at a shallow
angle of about 70˚ to the normal (almost skimming the air/water interface), what direction will
the laser beam have when it leaves the pool? We get:
 
1.33 Sin 70 1* Sin 
We need an angle which has a sine of about 1.25! We can’t do that. What will happen? All the
light will be reflected back into the pool. This is called total internal reflection. We can figure
out the angle at which this first happens, known as the critical angle, by looking at the incident
angle for which the refracted angle is 90˚ (the maximum). We’ll find
 
n2 Sin 90 
Sin c 
n1
n 
or c  Sin 1  2  for n1  n2
 n1 
We can take advantage of this effect by making a cable out of plastic with a high value of n and
then wrapping it with something with a lower value of n. This will make a “light pipe” or
optical fiber. If light strikes the sides at an angle greater than the critical angle, it gets bounced
back into the cable. Using this lets us guide light around corners, down pipes, etc.
More Polarization
When light strikes a nonmetallic surface and is reflected, the reflected wave will be polarized (at
least partially) parallel to the surface. For a particular angle, called Brewster’s angle, the
polarization is total. If the incident ray travels through a material n1 and is reflected from the
surface of a material n2, the angle needed for complete polarization is
n2
Tan  B 
n1
Dispersion
The index of refraction for a material is generally a function of frequency – in water, for
example, blue light moves slower than red light (the index of refraction is therefore higher for
higher frequencies). Different indices of refraction mean different angles of refraction (blue light
is bent more than red light by water). This is known as dispersion and it’s the reason for
rainbows and it’s also the way prisms break white light into its component colors. For water, the
speed difference between red and blue light is only about 1%, so the bending is slight.
Lenses
The purpose of a spherical mirror is to gather light and bend the reflected rays down to a point
(focus). We can use Snell’s law to see that we can do the same thing with refracted rays. Below,
look at the bundle of rays emitted by the candle. Without a lens or mirror, only a small fraction
of those rays will reach your eye.
Many of the rays will go over your head, hit your neck, miss you entirely, etc. We would like to
bend some of those rays so that they hit your eye instead of your shirt. We can do this with a
lens. It needs to be almost triangular at the top and bottom to bend light like a prism, and almost
flat at the center to avoid bending the light rays that will hit your eye anyway.
Now we have to do one more thing. All of the light that leaves the candle at a given instant
needs to arrive at your eye together (in time as well as space). It’s easy to see that the red &
purple rays above will have to travel a longer distance than the dotted black central ray. There’s
no way to speed up the red & purple rays, so we have to slow down the central ray. We can do
this by putting some glass in its path. For intermediate rays, we need to choose the shape so that
the light rays are bent the right amount and we need to choose the thickness so that they’re
slowed by the right amount. The shape that will do this (approximately) is the ordinary convex
lens (if it’s convex on both sides, it’s called double convex).
This lens is also called a converging lens since it makes (parallel) light rays converge to a single
point (known as the focal point and located at a distance called the focal length from the lens).
Parallel light going through the lens will look like this:
f
We can also have lenses which cause parallel light to spread out, or diverge. These are, of
course, known as diverging lenses. Formerly parallel light leaving a diverging lens won’t come
to a focus, but it will appear to have originated from a single point behind the lens. That point is
called the focal point, and its distance from the lens is called the focal length.
f
Lenses and Ray Tracing
As with mirrors, we have three rules for converging lenses (and 3 similar ones for diverging
lenses). They are (converging first): Draw a ray parallel to the principal axis until you hit the
lens, then draw a straight line through the focal point.
Principal
Axis
f
The next ray goes through the left-hand focal point, strikes the lens, and then starts traveling
parallel to the principal axis.
Principal
Axis
f
The third and final ray travels directly through the center of the lens without bending:
Principal
Axis
f
These three lines will intersect at a point, and the image will be displayed at this point.
Principal
Axis
f
For a diverging lens, the rules are a little different. The first ray is drawn parallel to the principal
axis, strikes the lens, and moves away from the axis in a way that makes it seem to have
originated at the focal point on the other side.
Principal
Axis
f
The second ray originally moves towards the focal point on the opposite side of the lens, but
strikes the lens and is redirected to be parallel to the principal axis.
Principal
Axis
f
The final ray goes directly through the center without bending
Principal
Axis
f
Finally, we see the image below (known as a virtual image – it can’t be projected. You need to
put your eye near the right end of the principal axis and look back to the left to see it)
Principal Axis
f
The diverging lens alone will always give a virtual, upright, and minified image. As before, we
can do this mathematically and avoid the problems of ray tracing.
Thin-Lens Equation and Magnification Equation
By some of the same geometry that we used in examining the connection between ray tracing
with mirrors and the equations for image locations and sizes, we can get the thin-lens equation
which is
1 1 1
 
do di f
where do is the distance from the lens to the object, di is the distance from the image to the object,
and f is the focal length of the lens (the distance from the lens where parallel light will come to a
point (converging) or appear to originate from (diverging)). Similarly, the magnification
equation is
m
hi
d
 i
ho
do
where m is the magnification and hi and ho are the image and object heights. The sign rules are
similar to what we saw with mirrors: f is positive for converging lenses and negative for
diverging lenses. Positive values of do mean the object is real (typically drawn on the left) while
negative values indicate a virtual object (typically on the right). Positive values of di give real
images (right side) and negative values mean virtual images (left). Positive magnifications are for
object and image in the same orientation while negative values mean they are inverted relative to
one another.
Optical Systems
We can combine two (or more) lenses and make various optical instruments. When one lens
feeds light to another lens, the image of the first lens acts as an object for the second lens. Some
general rules about converging lenses are also useful. We’ve already seen that parallel light
comes to a focus one focal length away from a converging lens. If the light is not parallel, it is
either converging or diverging when it hits the lens. Converging light will have a “head start” on
coming to a focus, so it will do so before it reaches the focal point. Diverging light is spreading
out and therefore will take longer to bring to a point, so it won’t happen until the light is more
than one focal length from the lens (if it happens at all – an object closer to the lens than one
focal length will produce light that is still diverging when it leaves the lens). Here are a few
examples, with the numbers included from the thin lens equation. For all pictures, assume the
lens has a focal length of 20 cm.
If we want to make a telescope, we’ll use two converging lenses – one (large) called the objective
(focal length fo) and a small one called the eyepiece (focal length fe). To understand how far apart
to place them, we just need to think about the situation for a minute. Light from distant objects is
effectively parallel. You can think about this geometrically, or you can calculate what 1/d0 would
be for something like the Moon, which is about 400,000,000 meters away. You’ll get a number
very close to zero, which we’d get for exactly parallel light. Also, the light leaving the telescope
should be parallel, because that’s the kind of light our eyes are most used to (since light from the
Sun is effectively parallel). Finally, the image produced by the objective will be the object for
the eyepiece. We’ve got two thin lens equations: for the first, let do approach infinity so that the
1/d0 term disappears. Our image will be at the focal length, so we’ll get
1 1

 d i  fo
di fo
which we already knew about parallel light. If we want parallel light to exit the eyepiece, that
means we should have di approach infinity (since parallel lines don’t cross after any finite
distance). If di is approximately infinite, the 1/di term disappears and we get (remembering that
we’re now looking at the eyepiece)
1 1

 d o  fe
do f e
So, our object needs to be one eyepiece focal length away from the eyepiece. Remember that the
objective’s image (which will be the eyepiece’s object) has to be one objective focal length from
it. This means the two lenses should be fo + fe apart. Parallel light comes in, and parallel light
goes back out.
Problems with a simple telescope like this include spherical aberration, or a failure of light rays
striking all parts of the lens to converge at exactly the same point (same as with spherical
mirrors) and chromatic aberration, which means different colors of light are focused to slightly
different points. Chromatic aberration happens because n depends slightly on the color of light
involved. This can be corrected (mostly) by using a pair of lenses – converging and diverging –
to bring the colors back to about the same point. Spherical aberration can be fixed by using
lenses with parabolic surfaces (expensive!) or by putting a “donut” in front of the lens to block
light that would hit far from the center of the lens. If the only light striking the lens is hitting
near the center, the rays will focus at approximately the same point. Fixing these effects is part
of the reason why a 35 mm camera lens really contains many lenses.
The eye is another complex optical system. Light entering your eye goes through several
different media on the way to your retina and will be refracted at each interface. Interestingly,
most of the bending occurs at the air-cornea interface because that’s the place where the
differences in n are largest. The actual lens of your eye doesn’t do much of the refraction, but the
part it does do is very important. Because you can’t really change the shape of your eye, you
have a fixed image distance (lens to retina). Using the thin-lens equation, if object distances are
changing as you look around, focal length of the lens is the only variable left. Muscles in your
eye change the shape of the lens and therefore change its focal length. When you’re looking at a
distant object, these muscles don’t have to do much work and therefore we find parallel light
easiest to observe.
If you’re nearsighted, parallel light will come to a focus before it reaches your retina. Since the
light rays are coming together too soon, the fix is to spread them out a little before allowing them
in your eye. For this reason, glasses for nearsighted people contain diverging lenses. When the
lens in your eye tries to focus the slightly-spreading light, it takes a little longer for the rays to
meet, and the image appears focused on your retina instead of in front of it.
If you’re farsighted, the lens of your eye can’t quite get the parallel rays bent to a focus by the
time they hit your retina. The fix, then, is to give the rays a little help by passing them through a
converging lens so that your eye can get them down to a point when they hit the retina.
How strong are your glasses? They are measured in diopters. This tells us about the focal
length of the lens in a simple way:
diopter 
1
f meters 
In other words, if you have glasses which are +2.5, they are converging (positive focal length)
and have a focal length of 0.40 m = 40 cm. One advantage to this way of measuring lenses is
that a larger number means a stronger lens (since it can bring parallel light down to a point in a
shorter distance). Nearsighted people will have lenses with negative diopters ( = diverging
lenses).
Interference
Just as was the case for sound waves, we can talk about the interaction between electromagnetic
waves by simple addition. If we want to know the magnitude of the electric or magnetic field
produced at a point by multiple sources, we just add the sizes of the individual fields. This is
known as the principle of linear superposition, and it leads to some interesting effects. The
red, green and blue waves below (different wavelengths and phases) combine to produce the
black wave.
We describe the way the waves will combine by looking at their wavelengths and phases.
Usually, we’ll look at sources producing the same wavelength of radiation, so phase differences
are really the only important things. For waves which are in phase (red and blue), the net effect
is to produce a wave of larger amplitude (black) (constructive interference)
If the waves are out of phase (red and blue), the net effect is to produce a flat line (destructive
interference)
To get stable interference between two sources, they must be coherent (meaning the phase
difference between the two sources doesn’t shift back and forth, but rather stays constant. A
common way to do this is to split the light from one source into two sources by means of a
couple of slits.) If the sources are coherent, the wave amplitude at a single place depends on the
difference in distances between the observation point and the two sources. If that difference is a
whole number of wavelengths, the observation point will receive a peak from one source at the
same time it receives a (different) peak from the other source (constructive interference). If the
difference is an odd number of half-wavelengths, a peak from one source will arrive at the same
time as a valley from the other source, giving us destructive interference.
Young’s Double-Slit Experiment
The effects of wave interference can be demonstrated using a screen which has two slits in it to
allow light to pass through. These slits need to be illuminated by coherent light of a single
wavelength coming from a single direction. We could do this with a laser now, but it was done
in the past by using a prism (or diffraction grating – more about them later) to direct different
wavelengths into different directions. Monochromatic light produced this way was directed onto
one slit (“A”) which served as a coherent source for the two slits (“B”) mentioned above
(coherent because the two slits were really being illuminated by a single source). Interference
patterns were then observed on yet another screen (“C”), as shown below:
A
B
C
We’ll get an alternating pattern of dark and light bands on screen “C”. We can understand which
parts will be light and which will be dark by looking at the difference in the path length between
a given point on the screen and the top slit vs. the path length between the same point on the
screen and the bottom slit (difference between blue and purple paths below). If that difference is
a whole number of wavelengths, we’ll get constructive interference as mentioned before. We can
use some geometry to understand what’s going on here.
A
B
 C
We notice that we can connect the top slit with the purple path using a line (green) which is
perpendicular to the purple path. This green line, along with the short part of the purple line and
the part of screen “B” between the slits, makes a right triangle. The hypoteneuse is the slit-to-slit
spacing, and the shortest side represents the difference in path lengths. We’ll call the angle
opposite to that shortest side (the angle at the top of the triangle) . If the spacing between slits
is d, the path length difference will be d Sin . Constructive interference appears whenever this
value is equal to a whole number of wavelengths, and destructive interference will result if this
distance is an odd number of half-wavelengths.  also can be connected to the position we’re
examining on screen C as shown below:

x

d
B
C
Notice that the position angle of the point we’re looking at is an angle  up the screen from the
center as measured by someone in between the slits. We can find this angle if we know the
distance between screens B & C and the height x shown above. Our general formula will then be
Sin   m

constructi ve OR
d

Sin    m 

1 

2 d
destructiv e
where m will be any whole number (0, 1, 2, …). The different values of m are known as orders
so that the m=2 bright spot is known as the second order bright fringe.
Thin-Film Interference
Interference will occur whenever we have two or more waves in the same place at the same time,
and we don’t need complicated arrangements of slits. When something like gasoline or oil is
spilled on the road and water is added, the lower-density oil will float on top of the water and
spread out into a thin covering for it. This gives us two interfaces between different media –
first, we have the air-oil interface, and below it, the oil-water interface. All three materials have
different values of n, so we expect some reflection and some refraction at each interface. The
reflections from the two interfaces can interfere with each other, and we get some interesting
effects.
t
We expect constructive interference when the extra distance traveled by the light going through
the gas/oil is equal to a whole number of wavelengths. There are just a couple of points to add
here: first, the wavelength we care about is not the wavelength of the light in a vacuum, but
rather the wavelength of the light in the gas/oil. This is only reasonable, since the extra distance
is traveled in the gas/oil, so the wavelength there is what should matter. The wavelength of light
in a medium with index of refraction n is related to its vacuum wavelength by
vacuum
medium 
nmedium
The extra distance traveled will just be twice the thickness of the film. The only other point to
consider is that reflected light can have its phase changed by 180˚ (or half a wavelength) when it
is reflected. Whether this happens or not depends on the indices of refraction of the two
materials at the interface: if n1 < n2 (where light starts in medium 1 and moves to medium 2),
there will be a phase change of 180˚ upon reflection. If n1 > n2, though, there is no phase change.
If we apply these rules to the picture above and we assume nair = 1, ngas = 1.4, and nwater = 1.33,
there will be a phase change going from air to gas, but not going from gas to water. This
introduces a half-wavelength difference between the two rays, even if twice the film thickness is
a whole number of wavelengths. Our rule in this case would then be
1
2 t  medium  m medium (constructi ve)
2
or
1
1


2 t  medium   m  medium  ( destructiv e)
2
2


where m is a whole number, as before. Notice that the half-wavelengths can be subtracted from
each side for destructive interference and we get the simple formula below:
2 t  m medium so t 
m medium
2
White light from the Sun contains all wavelengths of visible light. Different parts of the film
will have different thicknesses, meaning certain wavelengths are enhanced and others are
cancelled. That’s why we get a wide range of colors when a drop of oil hits a puddle of water.
Soap bubbles also have two interfaces – air outside to soap, and soap to air inside.
Diffraction
As was the case with sound, electromagnetic waves spread out after passing through a slit or
around a corner in a process called diffraction. This is the reason we can hear sounds around a
corner, and while we can’t usually see around a corner, we can see the spreading of light waves if
we have a small enough opening. We can understand this by realizing that each point on a wave
front acts like a source of new (spherical) waves. This is why we don’t get a very sharp shadow
from a small slit, as seen below:
Spread of light
with diffraction
New sources of
spherical waves
Path of light without diffraction
Wave fronts
strike slit
New wave
front
The purple wave front made from the addition of all the spherical red wave fronts produced at the
slit clearly spreads out. The amount of spreading depends on the ratio of the wavelength of
radiation to the width of the slit. For visible light ( = 400-700 nm) streaming through your
windows (W = 0.1-2.0 m), the spreading is insignificant and we see a fairly sharp edge to the
boundary between light and shadow on the opposite wall.
We can understand this by imagining the slit to be two adjacent slits W/2 in length. Different
points on the screen across from our slit will have different distances to the same part of the slit.
Likewise, different parts of the slit will have different distances to the same part of the screen.
For example, look at a point on the projection screen which is an angle  above the center of the
projection screen. Each different source in our slit will have a different distance to this point on
the projection screen, as shown below:

W
The similarities between this picture and the double-slit interference picture should be obvious.
If the path length difference between a point in the slit and the corresponding point ½ of a slit
width away, waves from those two points will cancel each other out. A point slightly below the
top will cancel out the contribution from a point slightly below the halfway point, etc. all the way
down. Therefore, the condition for destructive interference (darkness on the screen) is for the
path length difference between a point and the corresponding point ½ of a slit width below to be
½ of a wavelength.
W


Sin   m ( destructiv e) or Sin  m
2
2
W
What does this tell us about the point in the center of the projection screen (directly across from
the slit, where  = 0˚)? Because Sin 0˚ = 0, we can’t get destructive interference across from the
slit for any nonzero value of . Therefore, we get a bright spot directly across from the slit.
Resolving Power
When we look at two distant lights, we sometimes see them as a single light, depending on our
distance to them and their separation from one another. When we try to apply what we already
know to this situation to determine the ability of an optical instrument to separate the two sources
clearly, we say we’re measuring the resolving power of the instrument. These distant sources
will suffer diffraction when passed through the lens of our camera (or telescope, etc.). The math
for the case of circular openings instead of slits is a little different, but the general form is the
same. A circular aperture will produce a central bright spot surrounded by alternating circular
bright and dark interference fringes (this will look like a bulls-eye). The first dark fringe will be
found at an angle given by
Sin   1.22

D
where is the wavelength of the light and D is the diameter of the lens. Two nearby sources will
produce overlapping interference fringes on the film. The more they overlap, the harder it will be
for us to see them as two separate sources. It’s customary to use the Rayleigh criterion to
determine whether the two bulls-eyes can be adequately separated. This says we will be able to
see the two sources as two distinct objects when the first dark fringe of on falls on the central
bright fringe of the other. The angle between the two sources will clearly be small if we’re at the
limits of resolution, so we can use the small angle approximation which says that
Sin    for small 
Also, we must measure angles in radians for this to work. With this approximation, we see that
the angular separation between two objects must be at least min to resolve them as two separate
objects where min is given by
 min 1.22

D
This formula works for all wavelengths of radiation. It explains part of the reason why researchquality telescopes are usually very large: a large value of D (lens or mirror diameter) gives a low
value for min, meaning we can tell if a dot in the sky is a single star or a binary star system (very
common). Also, we can see that radio telescopes, which may have dish diameters of over 100m,
still have very bad resolution when used alone, since the wavelengths of light involved are in the
centimeter range themselves.
Diffraction Grating
If there are many parallel slits in a screen instead of just the two in our double-slit experiment,
we’ll have a device called a diffraction grating. These will produce bright areas on the
projection screen wherever the path lengths from a point on the screen to two neighboring slits
are a whole number of wavelengths different. For a diffraction grating with a slit separation of
D, the bright spots can be found where
Sin  m

for bright spots
D
We get an interesting result here if we shine white light on the diffraction grating. We have a
mixture of all colors (= all wavelengths) so we will get different values of  for each value of .
This means that different colors will appear at different places on the screen. This is telling us
that a diffraction grating will act like a prism, separating light by color! You can see this effect
for yourself if you take a CD and let sunlight reflect off of it. You’ll see different colors because
the tracks on the CD act like a diffraction grating.
We can even diffract X-rays if we use a really small value for D – something too small for us to
make, but that we can find in nature. The regular arrangement of atoms in a crystal (like salt)
produces a diffraction grating for X-rays. Conversely, we can use X-ray diffraction patterns to
figure out the structure of a crystal and the spacing of the atoms which make it up.