"It is manifest that everything in the world, whether it be substance or accident, produces rays in its own manner like a star... Everything that has actual existence in the world of the elements emits rays in every direction, which fill the whole world."
--Al-Kindi, De Aspectibus (ca. 860).
Preparation
Due 12:00 AM before start of lab
Pre-lab assignment 3 (*.html)
Equipment
rulers (1 m stick, 12" straightedge)
optical bench
f ≈ +12-13 cm converging lens (*.gif)
illuminated arrow box
image screen
lens holder
thin rod
legal paper
laboratory laptop
Microsoft Excel
Big Ideas
Converging lenses can create real images projected onto a screen.
Real images are inverted, but can either be enlarged or diminished with respect to the original object.
Real images are produced when object distances are greater than the focal length of the converging lens.
Ray tracings can be used to locate real images produced by (converging) lenses.
Data subject to experimental error can be estimated from the determining the range of values that produce a desired state, and represented graphically using error bars.
Goals
Students work in groups to observe formation of real images by a converging lens, in order to determine its focal length and model the image formation using a ray tracing.
Students learn how to handle data subject to experimental error from bracketing the range of lens-to-screen distances that result in a focused real image, calculate a representative percent uncertainty, and to display these using error bars on graphs.
Students build upon previous best practices to independently write an individual lab report, which can be submitted early, on time, or late depending on their personal initiative.
Tasks
(Optimally form groups of two students, three only if necessary.)
1. Experiment Set-up and Percent Uncertainty Trial
(Show calculations on worksheet to be checked-off; and to be included later in an individual, independent lab report.)- Mount an illuminated arrow box at one end of the optical bench; this will be the "object" for your converging lens. This position will not change for the remainder of the experiment.
- Clip your converging lens (approximate focal length f of +12 to +13 cm) to a lens holder, and mount this 35.0 cm away from your arrow box, using a meter stick to measure from the face of the arrow box to the center of the lens (ignore the markings on the optical bench, as it may not be calibrated to each mount). This is your object distance do, and because of uncertainties in the least count of the meter stick, wobbling of the mounts, and/or the thickness of the lens, we will assume a reasonable uncertainty of 0.5 cm for this measurement. Calculate the percent uncertainty of this sample object distance; this percent uncertainty will be assumed constant for all other data points (as this sample do value is near the mid-point of the data range):
Object distance percent uncertainty
= 100% × (0.5 cm)/(35.0 cm) = __________%.
- Mount a projection screen on the opposite side of the lens from the arrow box. Slide the screen back and forth until the (upside-down) image of the arrow box is more or less in focus on it; the resulting distance from the center of the lens to the projector screen (with an in-focus image) is the image distance di (again, measure this distance with a meter stick, and not with the markings on the optical bench). Because there is a qualitative range of values that the image could be considered in focus (due to different parts of the arrow box pattern not being able to be in focus at the same time, the clarity of the lens, and/or wrinkles in the projection screen paper), determine a reasonable minimum image distance and maximum image distance that the image could still be considered in focus:
Minimum in-focus image distance: __________ cm.
Maximum in-focus image distance: __________ cm.
- Calculate the mean value ("the average") of the minimum and maximum in-focus image distances, and their difference (maximum value minus minimum value):
Mean image distance = __________ cm.
Difference = __________ cm.
- Now calculate the percent uncertainty of this sample image distance; this percent uncertainty will be assumed constant for all other data points (as this sample di value is near the mid-point of the data range):
Image distance percent uncertainty
= 100% × (difference/2)/(mean image distance) = __________%.
2. Data Collection
(Show calculations on your own worksheet, to be checked-off; and to be included later in an individual, independent lab report.)- Develop an experimental linear trendline equation of how the inverse image distance (1/di) (dependent variable) depends on the inverse object distance (1/do) (independent variable). Refer to the example data table below to create your spreadsheet. Do not enter the "A..." column headings and "1..." row headings, as those are just spreadsheet "coordinates." Sample cell formulas to be entered below are highlighted in yellow.
For horizontal and vertical error bars, use the "Error Amount > Percentage" option to separately enter the horizontal and vertical percent uncertainty values previously determined from the sample data point above.
Object distance vs. image distance
| A | B | C | D | E | F |
| 1 |
Object distance |
Image distance |
Graph data |
| 2 |
do
(cm) |
min. di
(cm) |
max. di
(cm) |
mean di
(cm) |
(1/do)
(cm−1) |
(1/di)
(cm−1) |
| 3 |
15 |
|
|
=(B3+C3)/2 |
=1/A3 |
=1/D3 |
| 4 |
20.0 |
|
|
⋮ |
⋮ |
⋮ |
| 5 |
25.0 |
|
|
⋮ |
⋮ |
⋮ |
| 6 |
30.0 |
|
|
⋮ |
⋮ |
⋮ |
| 7 |
35.0 |
|
|
⋮ |
⋮ |
⋮ |
| 8 |
40.0 |
|
|
⋮ |
⋮ |
⋮ |
| 9 |
45.0 |
|
|
⋮ |
⋮ |
⋮ |
| 10 |
50.0 |
|
|
⋮ |
⋮ |
⋮ |
| 11 |
55.0 |
|
|
⋮ |
⋮ |
⋮ |
| 12 |
60.0 |
|
|
⋮ |
⋮ |
⋮ |
(Refer to the previous labs for instructions on how to generate a graph with independent and dependent variables with a linear trendline and error bars.)
- Print out one copy of your data table, and print out one copy of your graph (with trendline equation and error bars) for review by your instructor, who will check off this off for your in-class work. Then print out more data tables and graphs (and an *.xlsx spreadsheet transferred via USB drive, e-mail, cloud, etc.) for each person in your group to use to independently write an individual lab report to be turned in at the start of the next lab.
- Since this graph has an independent parameter of x = (1/do) and a dependent parameter of y = (1/di), then the thin lens equation can be expressed in terms of a linear equation:
(1/do) + (1/di) = (1/f),
x + y = (1/f),
y = −x + (1/f),
where the slope m of this trendline would be expected to be close to −1 (check to see if your graph has a negative slope!), while the vertical intercept b for the trendline should be the inverse focal length (1/f). Determine the experimental value of the focal length of your converging lens, with the appropriate number of significant figures (which should be two).
Converging lens focal length f = +__________ cm.
- Test the validity of the ray tracing model using your experimental converging lens focal length f, by drawing two separate ray tracings (use a 4 cm actual = 1 cm drawing scale, so they can fit on a page), assuming a scaled 4 cm object height (scaled down to 1 cm), and using these two object distances outside of your experimental data range:
- object distance do = +14 cm (scaled down to 3.5 cm).
- object distance do = +70 cm (scaled down to 17.5 cm).
Don't forget to scale the positions of the focal points by 4:1 as well! (You may instead use a different scaling factor (such as 5:1) as necessary to fit your ray tracing on a page, and/or tape several pages together.)
Measure the resulting ray tracing image distances, and scale them back up to the predicted image distances obtained for each case.
Comparison of predicted ray tracing and experimentally measured image distances
Object
distance |
Image distance |
Percent
error |
| predicted |
measured |
| +14 cm |
|
|
|
| +70 cm |
|
|
|
- Show your work in testing the validity of your ray tracing model by comparing the percent error between the predicted distances from the ray tracings and the measured experimental image distances. (These calculations will also be included in the conclusion of your independent lab report to be turned in during the next lab; this is for your instructor to check to see that you have taken all the necessary data in lab in order to write your report at home.)
- Documentation Rubric (task 2)
(Graded for the entire group)
| Score | Description |
| 3 | Sufficient amount of data points, graph/trendline and validation calculations complete, or very nearly so. |
2 | (No intermediate score possible.) |
| 1 | Substandard effort; insufficient data, problematic graph/trendline, validation calculations missing or incorrect. |
| 0 | Unacceptable or no significant effort. |
3. Independent, Individual Lab Report (checklist: (*.pdf))
(Due next lab)(You may either work on this during the rest of lab today, and/or later for homework.) Independently work on writing and complete an individual lab report, due next lab, which should include:- A descriptive abstract.
- Procedure (emphasis on materials used and how the experiment was set up (diagrams are okay), instead of step-by-step instructions).
- Data table, calculations and/or results (show work in calculating the focal length for the lens, your two ray tracings, and percent errors for the ray tracing and experimental image distances).
- Write out concluding statements regarding the validity of your ray tracing model by comparing the percent error between the predicted distances from the ray tracings and the measured experimental image distances. Include the specific relevant numbers in these statements, such that each can be read (and cited) on its own without referring to the above calculations and numbers.
(Refer to previous labs for suggested best-practice guidelines for each of these sections.) - Lab Report Rubric
(Due next lab; each student works on their individual write-up individually)
| Score | Description |
| 3 | (Essentially) complete, thorough, understandable, with very few or no corrections. |
| 2 | Minor problems; some corrections/revisions needed. |
| 1 | Minimally acceptable effort, essential/critical revisions needed. |
| 0 | Unacceptable or no significant effort beyond original experimental work. |
- Submission Modifiers
(Added/subtracted from lab report points)
| Modifiers | Description |
| +1 | Report is turned in "early" on the same day of data-taking; or in the first 10 minutes of the next lab. |
| 0 | Report turned in any other time during the next lab. |
| −1 | Report turned in the day after the next lab; up to one week late. |
| −2 | Report turned in more than one week late. |
(No negative net points are possible for a lab report; the lowest possible grade (after applying the submission modifiers) is zero.)
Follow-up
Complete this week's lab report and post-lab assignment, next week's pre-lab assignment, and review lab instructions.
Due 12:00 AM before start of next lab
Post-lab assignment 3 (*.html)
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