Geometric Optics
Teacher Notes
College Ready MSP
Experimental investigations and supporting activities are described that consider the properties
and interactions of both mechanical electromagnetic waves. This sequence of learning activities
addresses the following Arkansas Science Standards and Advanced Placement Learning
Objectives.
Arkansas Science Standards
WO.10.P.1Calculate the frequency and wavelength of electromagnetic radiation
WO.10.P.2 Apply the law of reflection for flat mirrors:
WO.10.P.3 Describe the images formed by flat mirrors
WO.10.P.4 Calculate distances and focal lengths for curved mirrors:
where
= object distance;
= image distance;
= radius of curvature
WO.10.P.5 Draw ray diagrams to find the image distance and magnification for curved mirrors
WO.10.P.6 Solve problems using Snell’s law:
WO.10.P.7 Calculate the index of refraction through various media using the following equation:
where
= index of refraction;
= speed of light in vacuum;
= speed of light in medium
WO.10.P.8 Use a ray diagram to find the position of an image produced by a lens
WO.10.P.9 Solve problems using the thin-lens equation:
where
= image distance;
= object distance;
= focal length
WO.10.P.10 Calculate the magnification of lenses:
where
= magnification;
= image height;
= object height;
= image distance;
= object distance
AP Physics Learning Objectives
The redesigned Algebra-Based Physics Exam will significantly extend and clarify expectations
within content about which the current Exam has largely emphasized disconnected facts and
procedures.
In the context of geometric optics the student is able to
• make claims using connections across concepts about the behavior of light as the wave travels
from one medium into another, as some is transmitted, some is reflected, and some is absorbed.
• make claims using connections across concepts about the behavior of light as the wave travels
from one medium into another, as some is transmitted, some is reflected, and some is absorbed.
• make predictions about the locations of object and image relative to the location of a reflecting
surface. The prediction should be based on the model of specular reflection with all angles
measured relative to the normal to the surface.
• describe models of light traveling across a boundary from one transparent material to another
when the speed of propagation changes, causing a change in the path of the light ray at the
boundary of the two media.
• plan data collection strategies as well as perform data analysis and evaluation of the evidence
for finding the relationship between the angle of incidence and the angle of refraction for light
crossing boundaries from one transparent material to another (Snell’s law).
• make claims and predictions about path changes for light traveling across a boundary from one
transparent material to another at non-normal angles resulting from changes in the speed of
propagation.
• use quantitative and qualitative representations and models to analyze situations and solve
problems about image formation occurring due to the refraction of light through thin lenses.
• plan data collection strategies, perform data analysis and evaluation of evidence, and refine
scientific questions about the formation of images due to refraction for thin lenses.
• make claims about the diffraction pattern produced when a wave passes through a small
opening, and to qualitatively apply the wave model to quantities that describe the generation of
a diffraction pattern when a wave passes through an opening whose dimensions are comparable
to the wavelength of the wave.
• qualitatively apply the wave model to quantities that describe the generation of interference
patterns to make predictions about interference patterns that form when waves pass through a
set of openings whose spacing and widths are small compared to the wavelength of the waves.
• predict and explain, using representations and models, the ability or inability of waves to
transfer energy around corners and behind obstacles in terms of the diffraction property of
waves in situations involving various kinds of wave phenomena, including sound and light.
In the context of E&M radiation the student is able to
• make qualitative comparisons of the wavelengths of types of electromagnetic radiation.
• describe representations and models of electromagnetic waves that explain the transmission of
energy when no medium is present.
Misconceptions Research
Studies alert us to the possible presence of the following misconceptions:
Different colors of light are different types of waves.
Light just is and has no origin.
Light is a particle.
Light is a mixture of particles and waves.
Light waves and radio waves are not the same thing.
In refraction, the characteristics of light change.
The speed of light never changes.
Rays and wave fronts are the same thing.
There is no interaction between light and matter.
Double slit interference shows light wave crest and troughs.
Light exits in the crest of a wave and dark in the trough.
In refraction, the frequency (color) of light changes.
Refraction is the bending of waves.
Prior Knowledge
It is assumed that students have completed the study of mechanics including simple harmonic
motion. They have developed skill in the construction and interpretation of graphs, the analysis
of data, the design of experiments, and mathematical reasoning involving angular coordinates.
Resources
Given the very large number of claims of understanding that can be assessed on the AP-2 Exam
a significant fraction of the class must devoted to this content. At least two weeks should be
given to activities involving lenses, mirrors and diffraction. Optical benches made from meter
sticks can be integrated with tasks involving the construction of representations. Diffraction
patterns are easily generated with laser pointers and a DVD. Diffraction patterns are also
generated by mounting a taut strand of hair in the beam of a stable laser. Thin and thick strands
produce measurable differences in the separation of anti-nodes.
Wave interference is well illustrated using a water ripple tank. If the old style of projection
systems was replaced by a Smart Board a video camera can be mounted above the tank surface
to display the feed from the camera. Interference is also illustrated with beats generated by metal
bars mounted on resonators.
Learning Sequence
The refraction at light at a boundary is familiar to your students. It is easily measured but the
physics is very difficult to understand. Without the attempt some important misconceptions go
unchallenged. One of these is that the interaction of radiation and matter can be explained in
terms of the elastic collision of ball at a plane surface. The conservation of momentum along the
plane is consistent with the “equal angles” law of reflection.
The particle model is abandoned entirely when we describe diffraction. The ripple tank and
analogy and the mathematics of a sum of sine waves provides a clear explanation of diffraction.
Between reflection and diffraction lies refraction. A simple representation of refraction can not
be based on either particles or waves. A mechanical model of a ball penetrating a water surface
does not tell us why the path is bent towards the normal.
A second misconception is that the speed of light is smaller when transmitted through a medium.
Some students will be excited by the idea that the light transmitted or reflected is actually reemitted by the atoms in the medium. But to use this to develop a model of reflection is
challenging and the subject of Feynman’s QED: The Strange Theory of Light and Matter. A
popular mechanical model based on the analogies in Feynman’s book is that the photons of are
held together by springs. Representing the atoms as children holding hands as they run towards
the water at an angle along the beach, their group trajectory is bent as the children who first enter
the water slow down. The result is that the group trajectory is bent towards the normal. While
this model is popular it embeds the misconception that light travels at different speeds and that
the photons are interacting. The difficulty provides an opportunity. Young’s explanation of
diffraction is understandable by algebra-based physics students. Feynman’s Cornell lectures,
available online, are also understandable for these students and provide a glimpse of quantum
effects.
A frequently used experimental approach to reflection is to use small plane mirrors glued to
blocks of wood to construct the virtual image behind the mirror. A directed-instruction lab is
included here that that supports that approach. If the learning objectives involve only
applications of the idea of refraction and not its explanation the labs included here will support
those objectives. A guided-inquiry is provided. Students who have thought about the reasons
for the steps they carry out in their study of reflection will be able to transfer the technique to the
method described here for refraction.
do
26
53
51.7
50.5
47.3
40.9
di
59
27
23.3
19.5
17.7
19.1
1/do
0.067
0.050
0.040
0.033
0.029
0.025
1/di
0.017
0.037
0.043
0.051
0.056
0.052
A structured inquiry is included that provides direct instruction
for data acquisition on an optical bench, but allows the student
to play a role in developing a strategy for the analysis. Typical
results are given in terms of the lengths shown in the figure
below. The focal length is found from the intercept f =1/0.085.
Finally, an approach to modeling diffraction is provided.
While the following pages might be appropriate to guide student work for reflection, refraction
and the analysis of images, no student work is provided for the investigation of diffraction. The
goal here is to challenge groups of students to develop a strategy for analysis. The story is
summarized in the following segment. The instructor can then provide guidance by letting the
story unfold through discussion with student groups. It is assumed that purple, green, and red
lasers can be made available for students to test their model.
Modeling Diffraction Patterns
• You have seen that the rays connecting wave interference maxima (antinodes) are spread apart
as the spacing between the sources gets smaller.
- Let x be the spacing between anti-nodes on the screen
- Let d be the spacing between the sources
-
The simplest model that describes your experience is x= a with a constant a
d
• You have seen that the geometry of the ray leading to an anti-node involves the tangent
- Let L be the distance from the source to the screen
- Let q be the angle between the ray and a bisector of the line between sources
-
The geometry tells you that tan(θ) = x
L
• You have seen that the geometry of the rays shows that anti-node separation decreases with
wavelength
- Let λ be the wavelength of the wave x = bλ
• You now have five variables at play: x, L, d, θ and λ .
• You have only three relationships connecting them; 5 unknowns and 3 equations is an incomplete
model
Here is how Young solved the problem. He reasoned that at the brightest spots there must be
constructive interference. So he drew two
waves coming from the two sources that were
in phase at the screen. The two waves have
different path lengths. The difference
between the longer and the shorter paths is the
red line segment. If the wave maxima at the
target are to coincide then the red line
segment must have an integer number of
wavelengths.
Putting It All Together
• You have seen that the rays connecting wave interference maxima (antinodes) are spread apart
as the spacing between the sources gets smaller.
x=
a
d
• You have seen that the geometry of the ray leading to an anti-node involves the tangent
tan(θ) =
x
L
• You have seen that the geometry of the rays shows that anti-node separation decreases with
wavelength
x = bλ
• The new piece of the puzzle is sin(θ) = nλ
d
• You now have four relationships connecting five variables … still not there!
What happens if the spacing between antinodes is very small compared with the distance to the
screen?
Use your calculators to evaluate x/L for the values measured. Then calculate θ. Now calculate
sin(θ). What useful (and very accurate) approximation results? Determine the spacing between
the slits in the diffraction grating using the model just obtained for all three wavelengths.
A Strategy for Finding Your Image
Imagine for a moment that you are 2.0 meters tall. A mirror hangs on the wall in front of you, its
top surface lined up with the top of your head. What is the minimum length of the mirror in
order for the image that you see of yourself to be complete?
In your notebook
Draw a stick figure picture of yourself facing a plane mirror. You see an image.
Where is the image? How big is the image?
Do the answers depend on the distance between you and the mirror?
Can you come up with a strategy that lets you answer these questions?
Draw the image of the stick figure.
The strategy is an
algorithm; a set of steps
that when followed lead to
a desired result.
The object whose image
you will find will be a
triangle that you draw at
one end of a sheet of long
paper.
You’ll create the image by
placing a plane mirror on the paper that lies on cardboard as shown above.
Make a line on the paper that shows the bottom edge of the mirror.
To find the image that you see (repeat for each vertex):
1. Place an object pin in a vertex of the triangle image.
2. Place a locator pin (Pin 2A in the picture) in the paper near the mirror and label it.
3. Find the line of sight that aligns the locator pin with the image of the object pin.
4. Put another locator pin along the line of sight and label it.
5. Remove the mirror.
6. Draw a line that extends beyond the plane of the
mirror connecting the locator pins
7. Replace the mirror.
8. Repeat steps 2 through 7 with a second pair of locator
pins. The image is at the intersection, as shown at the
left.
Now remove the mirror and connect the vertices of the
image triangle.
In your notebook:
Make measurements of the locations and properties of the image and object.
Draw a conclusion as rules about the image formed in a plane reflection.
Support your conclusion with evidence from your construction of an image.
Turn the paper so that the object is near you. Label the left and right sides of the object L and R.
Now turn the paper so that the image is near you. Label the left and right sides of the image L
and R.
In your notebook
Extend your rules to include a statement about the symmetry of a plane reflection.
The rays that you drew extending the line of sight through the mirror were virtual.
The path of a virtual ray can be located with a simple rule.
Using the protractor measure the angle that the line connecting two locator pins made with the
base line of the mirror. We’ll call this the reflected ray.
Draw a line connecting the location of the object pin at the first vertex of the object triangle with
the intersection of the baseline and the reflected ray. We’ll call this the incident ray.
Compare the angle between the incident ray and the line of the mirror with the angle between the
reflected ray and the line of the mirror. Do this process for the second vertex.
In your notebook
Extend your rules to include a statement about the angles
between an incident ray and a plane mirror and the
reflected ray and the plane mirror.
In your notebook
Draw the image at the left (or use the image constructed
in the lab) and make annotations on the drawing that
show where these rules have been obeyed.
Return to your stick figure drawing and revise it if
necessary.
The Image You See
The image of yourself that you see in the mirror is
formed according to some rules. If these mirrors
are violated then what you see may be disturbing.
Or it may be entertaining. The fun house mirror
breaks these rules:
• The distance of the image from the reflection
plane is the same as the distance of the object
from the reflection plane.
• The object is left-right inverted when the image
is formed.
• The angle of incidence of a ray from the object
to the reflection plane is equal to the angle of
reflection from the mirror plane to the eye.
In your notebook
Draw the image at the left (or use the image
constructed in the lab) and make annotations on the drawing that show where these rules have
been obeyed.
Return to your stick figure drawing and revise it if necessary.
Refraction of Light
When you look into the water at a fish, the ray of
light coming from the fish is bent when it crosses
the surface.
You think that the fish is not as deep as it really is.
What you see is the image of the fish.
Measurements of the locations of an object and its image can be made with a hemispherical dish
of water.
Put one locator pin at about the midpoint of the dish diameter.
Place an object pin beyond the curved surface.
Place a second locator pin so that the two
locator pins are aligned with the object pin
when viewed through the water.
Different positions of the object pin result in
different locations of the second locator pin.
Let Θincident be the angle between the surface normal and the incident ray from the object.
Let Θrefracted be the angle between the surface normal and the ray refracted as it crosses the
surface.
In your notebook
Write down the sequence of steps that you use to find the relationship between Θincident and
Θrefracted.
Draw a picture of your experiment.
Make a table to collect the measurements.
Using your calculator or a spreadsheet make a graph of the data.
Is a linear mathematical model a good representation of this relationship? If not, why not?
Using your calculator or a spreadsheet, test this model: sin(Θrefracted) = a sin(Θincident) for some
constant a.
Draw a conclusion comparing these two models.
Measurement of Focal Length
A. The focal length of a double convex single lens
Select a lens and estimate the focal length by looking at a distant object. Assemble the optical
bench, putting the lens on the bench at an object distance, do, larger than your estimated focal
length. Put the screen and light source on the bench as far apart as possible. If using a light bulb
place a screen with crossed arrows in front of the bulb. If using a candle the candle will be the
object and the screen with crossed arrows is not needed. Align the heights of all components.
The light source will either be a light bulb or a candle. The leads from the lamp are jacked into
the DC side of the power supply. The bulb has a small upper limit of current so before plugging
it in turn the power supply counterclockwise all the way. When increasing power to the bulb do
so slowly. The light does not need to be very bright.
Measure the size of the object, ho and record it.
Move the lens until the image seen on the back of the screen is most sharp. This will be an
interval. Measure the height of the image, hi. Record hi, do, and di. As a separate check on your
measurements, compare measured magnification (=di/do) with ratios of measured distances and
add this as a column to your data table.
Take at least five more data points by relocating the screen and then repositioning the lens until
the image is focused.
Using a spreadsheet to analyze the data graphically. Recall that the focal length satisfies
1 1 1
= +
f d o di
(1)
Repeat this process for a second double convex lens.
Your report includes data tables and the graphical calculation of the focal length.
Graphical analysis using equation (1) provides two estimates of the focal length, one estimate
from the x-intercept and one from the y-intercept. Your measurement of the focal lengths of
the two lenses must report the uncertainty.
B. The focal length of a compound lens
Eyeglass lens are concave on one side and convex on the other. The focal length of a double
concave lens can be measured by combining it with a convex lens. A ray trace in which the two
lenses are separated can guide your ray trace of the lenses when they are in contact. Complete
the ray trace before beginning this part of the lab.
Include this ray trace in your report. Remembering that distances are defined relative to
center of the lens, use this ray tracing to assess the following claim:
1 1 1
= +
f f1 f2
Based on your assessment, what assumption is being made in this “effective focal length”
equation? Include your answer in your report.
Using this relationship and the procedure used earlier to measure the focal length of the double
convex lens, f2, measure the focal length of the double concave lens, f1.
Repeat this process using the second double convex lens whose focal length was determined.
Your report must include data tables and graphical analyses of the focal lengths of the
compound lenses.
Include the uncertainty obtained in the measurement of each double convex lens and the
uncertainty obtained in the two measurements of f1 when you report the focal length of the
double convex lens.