"To find the number of pages in a one inch stack we could count them, but there is an easier way. Suppose we start our stack on page one and find that the last page in our one inch thick stack is page 538. We might at assume that there were 538 sheets in our stack, but we would be wrong if we did. Why? Because there are two numbers on each sheet, one on the front, and another on the back. So, if we start with 1 and end with page number 538, there are half as many sheets in the stack as there are page numbers, or 538/2 which equals 269."
--Robert A. Millikan, "How to Measure the Thickness of a Single Page
With Only a Twelve Inch Ruler,"
wbilljohnson.com/journal/math/pagethickness.htm (February 1, 2008).
Preparation
Pre-lab assignment 7 (*.html)
(Due 12:00 AM before start of lab)
Equipment
rulers/metersticks (12", 1 m, 2 m)
aluminum foil rolls
scissors
catalog/directory/book, with 500+ numbered pages (*.gif)
digital multimeter (with capacitance setting)
banana plug wires, alligator clips
two 1 kg masses
digital calipers
Big Ideas
A parallel-plate capacitor is constructed from two conductive sheets with the same area A, separated by a distance d, and its capacitance C is given by:
C = A/(4π⋅k⋅d).
Data subject to experimental error can be estimated from the determining the least count of measurement devices, and represented graphically using error bars.
Two different values can be compared by calculating percent differences.
Goals
Students work in groups to construct a capacitor and to measure its capacitance.
Students familiarize themselves with using digital multimeters to make measurements.
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 (Under construction.)
(Optimally form groups of two students, three only if necessary.)
1. Experiment Set-Up
(Show calculations on worksheet to be checked-off; and to be included later in an individual, independent lab report.)- Use a digital calipers to measure the overall thickness of all numbered pages in your book (e.g., pages 1 through 500, looking at only "top left" numbers). Note that the actual number of pages that you are measuring is one-half of the "numbered pages" (e.g., there are 250 pages with numbers 1-500). Then calculate the thickness of a single page.
"Numbered pages" = __________. (Should be an even number!)
Total thickness of "numbered pages" = __________ mm = __________ m.
Actual number of pages = __________.
Thickness of a single page = __________ mm = __________ m.
- Record the least count (the last significant figure) of your digital calipers; this will ultimately determine the uncertainty of the separation distance between capacitor plates (which use pages as spacers). Then calculate the percent uncertainty of your total thickness; this percent uncertainty will be assumed constant for all other data points.
Least count of digital caliper = __________ mm.
Separation distance percent uncertainty
= 100% × (least count (mm))/(total thickness (mm))
= __________%.
- Measure the height and width of a page in your book, and calculate the area.
Page height = __________ cm = __________ m.
Page width = __________ cm = __________ m.
Page area = __________ m2.
- Cut two pieces of aluminum foil the same size as the pages in your book, minimizing the amount that sticks out beyond the edges of the page. Insert these "plates" 20 numbered pages apart (that is, 10 actual pages between them) in your book, and slip in a small piece of foil for each aluminum "plate," to act as conductive leads to attach alligator clips. Then place two 1 kg masses onto the center of the book to apply a consistent amount of pressure to compress the pages.
- Plug wires into the "COM" socket and the capacitance socket of the multimeter (which may be labeled with a capacitor symbol, or "C," and/or "F"), and connect these wires to your capacitor plates. Turn the selection dial from "OFF" to "capacitance" (again, labeled with a capacitor symbol, or "C," and/or "F"). (After you are done making measurements for today's lab, be sure to turn the dial back to the "OFF" position, in case your multimeter does not have an automatic shut-off function.)
- Record the capacitance value from the multimeter (in nF), and note the least count of the display (which should be 0.001 nF or 0.01 nF--if the least significant figure is unsteady, then press the "RANGE" button to reduce the number of significant figures); this will be considered the uncertainty of your capacitance measurements.
"Numbered pages" = 20.
Actual number of pages = 10.
Capacitance reading = __________ ± _________ nF.
2. Analyzing Capacitance Dependence on Separation Distance
(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 capacitance C (dependent variable) depends on the inverse of the separation distance (1/d) (independent variable). Refer to the example data table below to create your spreadsheet; note that you are using your group's data from the previous table above, and data from four other groups. 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; cells to be copied and continued downwards are denoted with "⋮" symbols.
For the horizontal error bars, use the "Error Amount > Percentage" option to enter the percent uncertainty from your micrometer measurements, and for the vertical error bars, use the "Error Amount > Fixed Value" option to enter 0.001e−9 (the thousandths place from the meter readings of nF values, but may be different for your multimeter). Ctrl-click on the trendline equation and select "Format Trendline Label..." in order to change it to scientific notation and display more significant figures.
Capacitance vs. inverse plate separation distance
| A | B | C | D | E | F |
| 1 |
"Numbered
pages" |
Actual number
of pages |
Single page
thickness |
Inverse
separation |
Meter
reading |
Capacitance
value |
| 2 |
(unitless) |
(unitless) |
(m) |
1/d (m−1) |
C (nF) |
C (F) |
| 3 |
20 |
=A3/2 |
(from above) |
=1/(B3*C3) |
|
=E3*1e-9 |
| 4 |
24 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 5 |
28 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 6 |
32 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 7 |
36 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 8 |
40 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 9 |
50 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 10 |
60 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 11 |
80 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 12 |
100 |
⋮ |
⋮ |
⋮ |
|
⋮ |
| 13 |
200 |
⋮ |
⋮ |
⋮ |
|
⋮ |
(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/d) and a dependent parameter of y = C, then the capacitance can be expressed in terms of a linear equation:
C = A/(4π⋅k⋅d),
C = (A/(4π⋅k))⋅(1/d),
y = (A/(4π⋅k))⋅x + (0),
where the slope of this trendline would be expected to be A/(4π⋅k), (while the vertical intercept for the trendline would be expected to be zero). Knowing your value for the area of your capacitor foil "plates," determine the experimental value for the Coulomb's force constant k, with the appropriate number of significant figures (which is two or three?).
Experimental value for k = __________ N⋅m2/C2.
- On your individual worksheet (and your lab report), you should have two concluding statements using:
- the percent difference your experimental value for k (in paper) and the expected value of k = 8.99×109 N×m2/C2 (for vacuum):
% difference = 100 × ( |bigger| − |smaller| )/(average of the two absolute values),
where the bigger value would be reported as "_____% greater" than the smaller value. Note that percent difference is not the same thing as percent error (comparing a measured value with an expected value), and not the same thing as percent change (comparing how a measurement increases or decreases from an older value to a newer value), as you are just comparing k values in different materials (paper vs. vacuum). Interpret your result to conclude whether or not paper has a significantly different k value compared to the vacuum k value; and
- the linear trendline R-squared value. Interpret your result to conclude whether or not it is plausible that the capacitance of a capacitor depends on the inverse of the separation distance.
(Include these statements 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 difference in k values and the inverse dependence of capacitance on the plate separation distance. 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 7 (*.html)
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