Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Converging Lenses
Group G – Type 2 graphical analysis
By: Isaac Ilivicky & Noam Kantor
Purposes:
The purpose of this lab was to investigate the relationship between the object distance and the image
distance for real images produced by a converging thin lens. In order to do this, we specifically analyzed
the relationships between:


Image Distance vs. Object Distance
Image Height vs. Object distance
Hypotheses:


As the object distance increases, the image distance decreases.
As the object distance increases, the image height decreases.
Apparatus:







A converging thin lens
Multiple rails preferably with an attached scale (meter sticks on the sides)
A light source (light bulb)
The Object (an opaque sheet with a symbol of some sort cut out in the center, ours
happened to be an “L”)
A screen where the image will be shown (multiple sizes will most likely be necessary)
A computer with LoggerPro or an organized data table of some sort
Time, patience, and some humor
Procedure
1. Measure the focal length by pointing lens at an object very far away. The prime object
in our physics explorations has been a power pole across the highway. Adjust the
position of the screen until the object is very “in focus.” The distance from the lens to
the screen is the focal length of the lens.
2. In a dark laboratory, start with the object 15 focal lengths away form the lens. On the
opposite side of the lens from the object, move the screen until you find a well-focused
image. Record the distance from the lens is known as the object distance.
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
3. Change the object distance in increments of approximately one focal length (rounded to
the nearest 5.0 cm) until you reach a distance of 3 focal lengths away from the lens,
recording the image distance and image height each time. Our suggestion is that in
order to measure the same values without having to move the object every time (which
with our apparatus was part of the light source), it is easier to move the lens and the
screen.
4. When you are 3 focal lengths away from the lens, reduce the distance increment to 5.0
cm.
5. When you are 2 focal lengths away, reduce the distance increment to 1.0 cm until you
reach an object distance of 1.0 cm less than 1 focal length away.
6. Another reminder; at every position, remember to record the object distance, image
distance, and the image height.
7. Plot image distance vs. object distance and image height vs. object distance, and
proceed with the analysis.
Diagram:

Credit to the UCLA physics department, however, due to current situations, GO
TROJANS!
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Experiment 1 Mathematical Analysis
𝑑𝑖 → 𝑖𝑚𝑎𝑔𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓 → 𝑓𝑜𝑐𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑑𝑜 → 𝑜𝑏𝑗𝑒𝑐𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑑𝑖 − 𝑓 ∝
1
𝑑𝑜 − 𝑓
𝑑𝑖 − 𝑓 =
𝑘
𝑑𝑜 − 𝑓
𝑘=
∆(𝑑𝑖 − 𝑓)
1
∆(
)
𝑑𝑜 − 𝑓
𝑘 = 0.041𝑚2 𝑎𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑏𝑦 𝑙𝑜𝑔𝑔𝑒𝑟𝑝𝑟𝑜
𝑑𝑖 − 𝑓 =
0.041𝑚2
𝑑𝑜 − 𝑓
Type 2 Graphical Analysis
Once we plotted the image distance vs. object distance, we noticed an inverse proportion
relationship with asymptotes that were not on the axes of the graph. After further observation, we
realized that the two asymptotes were approximately equal to the pre-established focal length. And so,
we adjusted our data, subtracting the value of the focal length from the recorded data for the image
distances and the object distances, and then plotted this new set of data. Once this was plotted, our
proportional inverse relationship’s asymptotes were lined up with the axes of the graph. Once we
reached this point, we were able to linearlize the data, which resulted in a direct proportion when
plotting image distance vs. 1/object distance. A similar process was taken when analyzing the second
relationship, where we plotted image height vs. object distance. This graph also showed an inverse
proportion relationship, with only the vertical asymptote being off from the vertical axis by a value equal
to the focal length. After dealing with this phenomenon with the exact same process as before, we
were able to derive a direct proportion when plotting (using the adjusted data) image height vs.
1/object distance.
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Experiment 1 Error Analysis
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 = 0.038𝑚2 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 = 0.041𝑚2
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = |𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 − 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒|
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = |0.038𝑚2 − 0.041𝑚2 |
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = 0.003𝑚2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒
0.003𝑚2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =
0.038𝑚2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =7.9%
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Experiment 2 Mathematical Analysis
ℎ𝑖 → 𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑓 → 𝑓𝑜𝑐𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑑𝑜 → 𝑜𝑏𝑗𝑒𝑐𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
ℎ𝑖 ∝
1
𝑑𝑜 − 𝑓
ℎ𝑖 =
𝑘
𝑑𝑜 − 𝑓
𝑘=
∆(ℎ𝑖 )
1
∆(
)
𝑑𝑜 − 𝑓
𝑘 = 0.0042𝑚2 𝑎𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑏𝑦 𝑙𝑜𝑔𝑔𝑒𝑟𝑝𝑟𝑜
ℎ𝑖 =
0.0042𝑚2
𝑑𝑜 − 𝑓
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Experiment 2 Error Analysis
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 = 0.0038𝑚2 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 = 0.0041𝑚2
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = |𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 − 𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒|
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = |0.0038𝑚2 − 0.0041𝑚2 |
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟 = 0.0003𝑚2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝐸𝑟𝑟𝑜𝑟
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒
0.0003𝑚2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =
0.0038𝑚2
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟 =7.9%
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Error Explanation
Experiment 1: It was exceedingly difficult to tell whether
an image was focused at a given position when the image
distance was large because the image was dim anyways.
So it was hard to differentiate between dimness and
being out of focus.
Expeirment 2: The scale for the height only went to the
nearest 2mm, so there was a large amount of estimation.
At large object distances, the image was very small, and
so this small scale was difficult to read.
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
Final Mathematical Model
Experiment 1:
𝑑𝑖 − 𝑓 =
𝑓2
𝑑𝑜 − 𝑓
Experiment 2:
ℎ𝑖 =
ℎ𝑜 𝑓
𝑑𝑜 − 𝑓
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
𝑑𝑜 < 𝑓 → 𝑁𝑜 𝐼𝑚𝑎𝑔𝑒
Optics – Converging Lenses
Noam Kantor and Isaac Ilivicky
𝑑𝑜 ≈ 𝑓 𝑏𝑢𝑡 𝑑𝑜 > 𝑓 → 𝐸𝑛𝑙𝑎𝑟𝑔𝑒𝑑, 𝐼𝑛𝑣𝑒𝑟𝑡𝑒𝑑 𝐼𝑚𝑎𝑔𝑒
𝑑𝑜 ≫ 𝑓 → 𝑅𝑒𝑑𝑢𝑐𝑒𝑑, 𝐼𝑛𝑣𝑒𝑟𝑡𝑒𝑑 𝐼𝑚𝑎𝑔𝑒