Physics 112

9 April 2024
Optics Lab

Theorist Experimentalist Analyst Communicator

Kayla Vallejo Ella Purvis Jessalyn Swanson Eva Rickard

Theorist:
Background and Theory

Week One
In order for us to determine the relationship between how the height of the object is affected by
the distance from the object we built pinhole cameras. We hypothesized that the equation
hi d i
= is the correct equation for determining magnification. We assume that the light stays in
ho d o
the same position and that the holes in the cameras are the same size. We predict that the
distances will vary between the two cameras. A concept that is relevant is that a camera works
by allowing a light to go through a hole to project an image onto a screen. The motivation of this
lab is to be able to understand how light entering a hole can project an image based on its
magnification.

Week Two
We were trying to determine the values of the length of the focal lens. We hypothesized that the
1 1 1
equation = + is not going to be proven false as the thin lens equation. We assume that
f do di
none of our lenses have smudges and that the lens, object and image will not move after our
measurements are taken. We predict that we will be able to find the focal lens point.
Experimentalist - Procedures and Data:
Week 1
In order to begin our experiment, we had to assemble a cardboard box, while one already came
pre-assembled. We cut a small hole in one end and a large rectangle in the other end. Then we taped a
piece of tinfoil over the small hole, poked a tiny hole in the middle, and then put transfer paper over the
large hole. Label the preassembled camera 1 and the other camera 2.
Next, we turned on an LED light in the shape of an F and positioned the tin foil side of our
cameras toward the light and moved the cameras around till an image appears on the transfer paper.
We gathered the following observations:

Observations Height of Distance of Length of Height of LED

image Camera and camera F
LED
 Camera 1 Small upside 2.0cm +/- 46cm 19.7cm+/- 3.7cm+/-
down F, more 0.05cm +/-0.5cm 0.05cm 0.05cm
defined circles
Camera 2 Big upside 4.0cm+/- 36cm+/-0.5cm 56cm+/-0.5cm 3.7cm+/-
down F 0.05cm 0.05cm
From this table we can observe that the smaller the camera, the smaller the projected image.
For the next activity set up the following objects:

Our measurements are as follows:

Do = 43cm +/- 0.5cm
Ho = 3.7cm +/- 0.05cm

For this experiment we changed the distance of Di to see how it would change Hi (the height of the
projected image in the screen). Our measurements are shown in the following graph:

4
Di vs Hi
3.5
f(x) = 0.082 x + 0.36
3
Height (cm)

2.5
2
1.5
1
0.5
0
15 20 25 30 35 40 45

Distance (cm)

Our graph also has errors bars of 0.05 cm for the height and 0.5cm for the distance, though they are hard
to see on the graph.
Hi Di
Using the slope of our graph we were able to derive the magnification equation: =
Ho Do
Once we have this magnification equation, we predicted the distances we would have to place both
cameras (Do) in order for the image to be the same size:
Camera 1: 36.45 cm +/- 1.496 cm
Camera 2: 66.60 cm +/- 2.658 cm
For activity 3 we created an aperture by adding a new piece of tin foil over the hole of one camera and
adding a second one only taped on one side like a flap. On the bottom piece, poke a bigger hole and on
the top piece, poke a tiny hole.
Using our apertures we gathered the following information:
Flap Closed: Hi = 2.0cm +/- 0.05cm with very crisp circles and image
Flap open: Hi = 2.0 cm +/- 0.05cm with super undefined circles and blurry image
Variables: All measurements are in centimeters (cm).
Week 2
For this week, start by setting up the following objects using one of the convex lenses:

Move the lens around until an image forms on the screen.

We collected the following data and we were able to calculate the focal length using the thin lens
equation:
Di = 13.5cm +/- 0.05cm
Do = 88.0cm +/- 0.05cm
F = 11.7cm +/- 0.0615cm
We then repeated this process for the second convex lens and got the following data:
Di = 77.0cm +/- 0.06cm
Do = 23.0 cm +/- 0.05cm
F = 17.71cm +/- 0.0677

For the next activity of this week we replaced one of the convex lenses with a concave lens and
discovered that the image is projected on the lens and not on a screen.
We then created the new setup:
Move the lenses around until a clear image is on the screen. We collected the following measurements:
F2 (convex lens) = 11.7cm +/- 0.05cm
Do1 = 25 cm +/- 0.5cm
Di2 = 39cm +/- 0.5cm
Change of d = 5cm +/- 0.5cm
We then used the thin lens equation to find Do2 (where the object is for lens 2).
Do2 = 16.71cm +/- 0.642 cm
From here we know that Di2 is the same as Do2 minus the distance
Di2 = 11.71 cm +/- 1.142 cm
We used the thin lens equation one last time to find the focal length of F1 (concave lens):
F1 = 7.97 cm +/- 1.294 cm
Variables: All measurements are in centimeters (cm).

Analysis:
Week 1:
The goal of the week 1 experiment was to develop a hypothesis and model, estimate uncertainty, and
identify the scope of the investigation. We did this by building two different pinhole cameras and taking
the dimension to and using our graph line determine the formula for slope which we determine to be: y=
0.082x+ 0.36.
hi d i
We then used this slope equation to determine our magnification equation: = . Assuming
ho d o
the light stayed in the same position and the holes in the two cameras were the same size, we took this
magnification equation and used it to calculate which distances these pinhole cameras would project the
same image size. We set the light image to display at 2 cm tall. We hypothesized that the division
uncertainty propagation would be the accurate equation to calculate our uncertainty values.
Distance Calculations:

Constants: h o = 3.7 cm hi =2.0 cm

Smaller Box:
di 19 cm
do= → do= =36.45 cm

( )
hi
ho ( 2.0 cm
3.7 cm) away from light

Bigger box:
di 36 cm
do= → do= =66.60 cm

( )
hi
ho ( 2.0 cm
3.7 cm) away from light

We then took this equation and calculations to determine our uncertainty equation and values using the
equation:
δ d o δ di δ ho δ hi
= + +
do di ho hi
Small Box Uncertainty:

( )
δ d o δ di δ ho δ hi 0.05 cm 0.05 cm 0.05 cm
= + + → δ d o= + + ⋅36.45=± 1.496 cm
do di ho hi 19.7 cm 3.7 cm 2.0 cm
Large Box Uncertainty:

( )
δ d o δ di δ ho δ hi 0.05 cm 0.05 cm 0.05 cm
= + + → δ d o= + + ⋅66.60=± 2.658 cm
do di ho hi 36 cm 2.0 cm 3.7 cm
After performing these calculations, we tested our distances, and saw that at the calculated distance for
each box, the image was 2 cm tall, showing that our calculations were accurate even when
disregarding the uncertainty values. Our hypothesis that this was the accurate uncertainty propagation is
supported.
Week 2:
The goal of week 2 experiment was to test a given hypothesis, develop a model, estimate uncertainty, and
identify the scope of investigation with thin lenses and focal length. We were testing the difference
between two different convex lenses and finding their focal length. We were given a hypothesis for this
1 1 1
experiment which was the thin lens equation: = + . We then ran our experiment
f do di
and took measurements for the lenses. For lens 1,
1 1 1
we calculated = + → f =11.70 cmand did the same for lens 2 getting f =17.71 cm. We
f 88 cm 13.5 cm
used these values to calculate uncertainty for the measurement using the equation:
δf =
( di do
+
d i +d o )
δ d i δ d o δ d i+ δ d i
+ so lens 1 uncertainty was:

δf = ( 0.05 +
13.5 cm 88 cm
+
13.5 cm+88 cm )
cm 0.05 cm 0.05 cm+0.05 cm
⋅11.70cm → δ f =± 0.0615 mm and we 1

did the same for lens 2 getting δ f 2=± 0.0677 mm .

Focal lengths for lens 1 and 2:

f 1=117.0± 0.0615 mm f 2=177.1± 0.0677 mm
For the next experiment we tested both convex and a concave lens the same time, one being stationary,
and the other at a distance needed to be calculated by the thin lens equation which we did as follows:
1 1 1 1 1 1
= − → = − → d o 2=16.71 cm
d o 2 f 2 di 2 d o 2 11.7 cm 39 cm
which is the distance we placed lens 2 away from lens 1 to get the results we needed. We further used the
equation to find the distance between lens 1 and the image resulting in d i 1=11.71 cm . Using
this value, we were able calculate the focal length for lens 1 and its image getting f 1=7.97 cm . For this
we are then given an uncertainty propagation expression to calculate uncertainty for our di and f1. The

equation was given as follows: δ f 1=f 1

[ δ d o 1 δ di 1 δ d o 1+ δ d i 1
do1
+
di 1
+
d o 1+ d i 1 ]
We calculated our uncertainties to be:
δ f 1=± 1.294 cm , δ d o 1=±0.0642 cm ,∧δ d i 1=± 1.142cm .
The focal lengths given for the lens were 100 nm and 200 nm, but for the focal length we were testing, we
used the 100 nm lens. Our calculated focal length for the 100 nm lens was 1 117.0 +/- 0.0615 mm and
lens 2 was 177.1 +/- 0.0677 mm. Although our lens calculations were not withing the uncertainty range of
the given value the goal of the lab was to calculate more accurate focal lengths as the given are slightly
off from their true values. Our hypothesis was supported as the uncertainty values aligned with the
equation we were testing.
This specific lab taught us about thin lenses and how they apply to our eyes and giving us understanding
of how we see. This lab was important as it gave us insight into how corrective lenses work and how
specific glasses work for a person with vision problems.

Conclusion (Eva Rickard)

WEEK 1
hi d i
We hypothesized the equation = for the determination of magnification. After our experiment and
ho d o
analyzing our results, we concluded that our equation held merit even when factoring in the uncertainty of
our results as well. As far as the size of the uncertainty, they were small compared to our results for the
distance, but also could stand to be reduced father if the experiment were repeated.
As far as assumptions go, in the first week, we assumed that the light would not shift position and that the
holes poked into the cameras were the same size (though realistically we do not believe them to be
definitively identical for this lab).
As for predictions, we predicted that the distances between the two cameras would vary as their sizes are
different and we believed that would affect the distance, this turned out to be supported by our data as the
relative distances between the two cameras varied even when adding in the uncertainty.
Our motivation this week came from the desire to understand more about the laws of magnification, as
well as to understand more about the inner working mechanisms of cameras. Knowing these and being
able to apply our understanding could lead to us being more knowledgeable photographers and just more
informed consumers.

WEEK 2
For week two, we were trying to determine the values of the length of the focal lens. Here, we
hypothesized that the thin lens equation would give us accurate data and that the equation would not be
proven false.
For assumptions: we assume that the lenses are not harmed and un-smudged in any significant way that
would impact our results, as well as that the object and image will not move after our measurements are
taken.
We predicted that we would be able to find the objects' focal point.
Although the results from our experiment this week did not fall within the range of given values, the goal
of the lab was to apply our knowledge and attempt to find more accurate focal lengths as the given were
slightly off their true value. We concluded that our hypothesis was supported as the uncertainty values
aligned with the equation we were testing.
This week was relevant to our everyday lives because, at some point in our lives, most of us will have to
wear corrective eyewear or love someone who does. In fact, two-thirds of American adults wear some
kind of corrective eyewear (Washington Post, 2023). This statistic shows the relevancy to our everyday
lives because, at some point in our lives, most of us will have to wear corrective eyewear or love someone
who does. To become better people and partners, a greater understanding of the application of lenses and
their effects is needed. By knowing the physics behind diagnosing near and farsightedness, we are
becoming more informed citizens.