"When studying wave interference, one often wants to know the difference in path length for two waves arriving at a common point P but coming from adjacent sources. For example, in many contexts interference maxima occur where this path-length difference is an integer multiple of the wavelength. The standard approximation for the path-length difference [Δℓ] is:
Δℓ ≈ d⋅sinθ,
where d is the distance between the sources, and θ is the angle [as measured from the middle line between the sources]. A common derivation [of this equation] begins with the seemingly paradoxical approximation that two paths that meet at a common point can be treated as parallel...
"Students are often confused by that approximation--how can it be correct to treat the rays as parallel if they converge at point P? If they aren't exactly parallel, then how much error is involved in using the approximation? ...As a result, students may opt to accept [this equation] passively rather than develop any sense of ownership of the equation."
--Seth Hopper and John Howell, "An Exact Algebraic Evaluation
of Path-Length Difference for Two-Source Interference,"
The Physics Teacher, vol. 44 no. 8 (November 2006), p. 516.
Preparation
Pre-lab assignment 5 (*.html)
(Due 12:00 AM before start of lab)
Equipment
rulers/metersticks (12", 1 m, 2 m)
large laminated cardstock protractor
36" wide butcher paper
masking tape
λ = 2.0 cm "wave strips" (*.pdf)
Big Ideas
Two adjacent (coherent) in-phase sources will produce constructive interference spots (maxima) on a screen when their path length difference is an integer multiple of the wavelength:
Δℓ = | ℓ2 − ℓ1 | = m⋅λ.
If the adjacent sources are spaced a distance d apart, then the path length difference can be approximated by:
Δℓ = d⋅sinθ.
where the angle θ is measured from the middle line between the sources.
Data subject to experimental error can be estimated from calculating discrepancies from another independent method of measurement, and represented graphically using error bars.
Goals
Students work in groups to visually observe how two adjacent sources can produce constructive interference spots (maxima) on a screen, and how well path-length differences can be approximated for two non-parallel paths by d⋅sinθ.
Students learn how to handle data subject to experimental error, 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 Visualizing Constructive Interference Spots ("Maxima")
(Show calculations on worksheet to be checked-off; and to be included later in an individual, independent lab report.)- Cover your entire lab table with butcher paper, creasing and folding it down over the long edges, and securely taping it such that it does not shift. Measure a mark 30 cm from a short end of your table, and draw a pencil line across the butcher paper such that it perpendicular to the long edge of the table. This is the midline of your two adjacent sources.
- Along the long edge of the table, make a mark 7.5 cm on either side of the midline. These will be the locations of your two in-phase sources that will both send waves to the other long edge of the table and constructively interference at certain locations there.
- Have one student firmly hold the end of each wave strip on the start of these two source locations at one end of the long edge of the table, but loose enough such they can independently pivot and meet at the midline on the other side of the long edge of the table. Verify that these two waves will constructively interfere when they meet at the midline location.
- Have another student gently move the "free" ends of both wave strips such that they meet at another location down along the long edge of the table, away from the midline location. Eventually the two waves will eventually constructively interfere where they meet at the long edge of the table. Mark this location, and repeat by moving the "meeting point" of these two wave strips further and further down the long edge of the table.
- For each constructive interference "spot" along the long edge of the table, separately measure the path length ℓ1 (from one source to the constructive interference "spot") and then measure path length ℓ2 (from the other source to the constructive interference "spot"). Fill in the data table below. (If your wave strips are too short to locate the last maximum, then you can skip it.)
Theoretical vs. experimental path length differences
| A | B | C | D | E | F |
| 1 |
"Maxima
number" |
Theo. path
length diff. |
Measured path lengths |
Exp. path
length diff. |
Error |
| 2 |
m
(unitless) |
m⋅λ
(cm) |
ℓ1
(cm) |
ℓ2
(cm) |
| ℓ2 − ℓ1 |
(cm) |
(cm) |
| 3 |
0 |
0.0 |
|
|
=ABS(D3-C3) |
=ABS(E3-B3) |
| 4 |
1 |
2.0 |
|
|
⋮ |
⋮ |
| 5 |
2 |
4.0 |
|
|
⋮ |
⋮ |
| 6 |
3 |
6.0 |
|
|
⋮ |
⋮ |
| 7 |
4 |
8.0 |
|
|
⋮ |
⋮ |
| 8 |
5 |
10.0 |
|
|
⋮ |
⋮ |
| 9 |
6 |
12.0 |
|
|
⋮ |
⋮ |
| 10 |
(7) |
14.0 |
|
|
⋮ |
⋮ |
- Ideally the experimental path length differences for each constructive interference spot should be exactly integer multiples of the wavelength (2.0 cm), but realistically there will be some errors due to the exact placement/pivoting of the wave strips on their sources, and eyeballing the constructive interference locations where the wave strips overlap. You should relocate/remeasure any constructive interference spot and path lengths that you feel needs improvement; log your old original (large error) measurements below (if any) before updating your data table with your improved measurements.
Documentation of discarded measurements (if any):
2. Validation of the d⋅sinθ Approximation
(Show calculations on your own worksheet, to be checked-off; and to be included later in an individual, independent lab report.)- Using a 2 m stick draw lines from the midpoint between the two sources to each of the constructive interference spots that you have drawn. Measure the angles for each of these lines using the large laminated cardstock protractor to the nearest 0.5°, where the midline is defined to have an angle θ = 0°.
- Develop an experimental linear trendline equation of how the path length difference approximation d⋅sinθ (dependent variable) depends on the experimental path length difference | ℓ2 − ℓ1 | (independent variable). Refer to the example data table below to create your spreadsheet; note that you are using data from the previous table above. 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 the horizontal error bars, use the "Error Amount > Custom > Specify Value" option to enter the column of error values previously determined (and repeated below), and for the vertical error bars, use the "Error Amount > Fixed Value" option to enter 0.13 (as (15.0 cm)⋅sin(0.5°) = 0.130898... cm, or 0.13 cm to two significant figures) for the positive and the negative error bar values.
Approximation vs. experimental path length differences
| A | B | C | D | E | F |
| 1 |
"Maxima
number" |
Exp. path
length diff. |
Error |
Maxima
angle |
Approx. path
length diff. |
Approximation
uncertainty |
| 2 |
m
(unitless) |
| ℓ2 − ℓ1 |
(cm) |
(cm) |
θ
(°) |
d⋅sinθ
(cm) |
d⋅sin(0.5°)
(cm) |
| 3 |
0 |
|
|
|
=15*SIN(PI()*D3/180) |
0.13 |
| 4 |
1 |
|
|
|
⋮ |
0.13 |
| 5 |
2 |
|
|
|
⋮ |
0.13 |
| 6 |
3 |
|
|
|
⋮ |
0.13 |
| 7 |
4 |
|
|
|
⋮ |
0.13 |
| 8 |
5 |
|
|
|
⋮ |
0.13 |
| 9 |
6 |
|
|
|
⋮ |
0.13 |
| 10 |
(7) |
|
|
|
⋮ |
0.13 |
(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 = | ℓ2 − ℓ1 | and a dependent parameter of y = d⋅sinθ, then the path length difference approximation can be expressed in terms of a linear equation:
| ℓ2 − ℓ1 | = d⋅sinθ,
(+1)x + 0 = y,
where the slope m of this trendline would be expected to be +1 (while the vertical intercept b for the trendline would be expected to be zero), if this approximation were exact. Record the slope value, which is the proportionality constant between the experimental and approximate path length differences.
Slope = +__________.
- On your individual worksheet, show your work in testing the validity of the path length difference approximation by comparing the percent error between your trendline slope and the expected slope of +1. Interpret this result as whether the path length difference approximation is slightly lower or higher than the actual experimental path length differences that were measured from each source. (This calculation 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.
- Write out concluding statements regarding the validity of the path length difference approximation by comparing the percent error between it and the actual experimental path length differences that were measured from each source. 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 5 (*.html)
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