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A playful exploration of the extensive property of resistance and the intensive property of resistivity.
1
1.1
OBJECTIVES
PREREQUISITE SKILLS AND KNOWLEDGE
Students should be competent in algebra, have experience reading and interpreting circuit diagrams, using
LabVIEW, propagating error, and using Excel and R to graph and analyze data.
1.2
RESEARCH SKILLS
After this lab, students will have had practice in:
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1.3
following laboratory protocols
using a laboratory notebook
building electrical circuits
building circuits from circuit diagrams
using LabVIEW to control and collect data from sensors
using a Vernier scale caliper
measuring current and voltage
using Excel to propagate error
organizing experimental data
using Excel to analyze experimental data
using R to graphically analyze experimental data
error analysis using R
LEARNING OBJECTIVES
After this lab, students will be able to:
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Describe the relationship between voltage, current and resistance in circuits that obey Ohm’s Law
Translate between a simple circuit diagram and a circuit
Differentiate between resistance and resistivity
Relate a current to the motion of charged particles
2
PRE-EXPERIMENT
By now you are familiar with the role of a resistor in a circuit. In the next two experiments you will use
Play-Doh to explore the effect of the resistor’s size and shape on its electrical resistance.
2.1
RESISTANCE
The resistance of an object is described by the equation, V=IR, where V is the voltage difference across
the object and I is the current through it.
For a given voltage, how does increasing the resistance affect the current?
A resistor can be made from any kind of material; even metals have some resistance. For a given material
you can make the resistor into many different kinds of shapes.
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2.2
RESISTIVITY
Resistance of a material depends on the size and shape of the resistor. The resistance also depends on the
material the resistor is made up of.
Why would the resistance depend on the material the resistor is made up of?
The resistivity  of a material is defined by its relationship to resistance R,
(1)
ρ
In the equation, the resistivity is indicated by the symbol ρ (rho), the variable L is the length of the resistor
and A is the cross-sectional area. The SI unit of resistivity is the ohm meter ( m). In this series of
experiments you will first measure the resistivity of Play-Doh and then examine the effect of shape on
resistance.
The cylindrical resistors shown below are made of the same material. Use Equation (1) to predict which
would have a greater resistance.
2.3
MEASUREMENT WITH VERNIER CALIPERS
You will be using calipers with a Vernier scale to measure the dimensions of the objects in this lab. The
Vernier scale provides an additional significant digit in your measurements. There are a number of on-line
tutorials on Vernier scales and calipers. Here are links to two:*
http://hyperphysics.phy-astr.gsu.edu/hbase/class/phscilab/vernier.html
http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html
*
Note that both of these tutorials, and many others you will find on the web, have Java tutorials. If you can get them
to work, that’s great, but you can probably get enough of an idea of what is going on from the words and pictures.
Look for some YouTube videos, if you find videos more helpful.
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In the example image above, the diameter of the acrylic sphere is read as 1.280 ± 0.002 cm. The zero line
(white arrow) points to a measurement between 1.2 cm and 1.3 cm. Estimation might get you to 1.27 or
1.28 cm, but the Vernier scale allows you one more significant figure. Look for the spot where a line from
the upper scale exactly meets a line from the lower scale. In this example, that meeting takes place at
1.280 cm (orange box). If the line just to the right of the 8 had been the place where the two lines joined
exactly, then you would record 1.282 cm. The last digit is always open to dispute, so the measurement
would be reported as 1.280 ± 0.002 cm. Why 0.002 cm?
2.4
MEASUREMENT ERROR AND CALCULATIONS
2.4.1
Standard Deviation
The standard deviation is the average deviation of a group of values about a mean. For example, in this
experiment you will be measuring the diameter d of a Play-Doh cylinder at five different points along its
length and then calculating the area A at each diameter. To calculate the average area you will add up the
measured values (Ai) and divide by the number of measurements (N):
̅
(1)
∑
To calculate the error in this value, use the standard deviation :
(2)
∑
̅
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2.4.1.1 Practice using Excel to Calculate Average and Standard Deviation
You can use Excel to calculate the average cross-sectional area and the standard deviation in the crosssectional area for the five recorded diameter measurements in cm: 3.110, 3.285, 3.000, 3.195,3.098,
assuming an estimated error of 0.002 cm for each measurement.
1. Label your data by entering descriptive text into cell A2.
2. Enter your measured data into a row or column extending from the label. Do not leave blank cells
between populated cells. They will be read as zeros.
3. Label the row or column for the calculated area.
4. In the row or column following the first datum, enter the formula for the area. In this example, the
formula would be written =PI()*(B2/2)^2.
5. Copy the formula into the remaining cells.
6. Type the labels for your average area and the standard deviation in your average area.
7. Next to the label for the average area, use the AVERAGE() function to compute the average area.
(Use Google to look this up if you don’t remember how to do it).
8. Next to the label for standard deviation, use the STDEV.S() to compute the standard deviation.
You may have noticed that Excel offered two kinds of standard deviation, population (STDEV.P) and
sample (STDEV.S). If you are measuring data for an entire population, for example, the heights of all
students in a class, then you would use the population standard deviation. However, if you are collecting
only a sample of the entire population, for example, a random sample of the heights of all the students at
UF, then you would use the sample standard deviation.
2.4.2
Propagating Uncertainties Redux
For this lab, you will need to propagate error through calculations that include multiplication and division.
Recall the form of the equation for addition/subtraction below:
(3)
And here are the error propagation formulas for multiplication and division, which you will also use for
this lab.
(4)
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(5)
2.4.3
Propagated Uncertainties in Resistance and Resistivity
For this experiment, you will need to propagate the uncertainties in the calculated values of resistance and
resistivity.
2.4.3.1 Uncertainty in the Resistance
The formula for the resistance is
.
1. Create columns in Excel for the Voltage (V), Uncertainty in the Voltage (V), Current (A),
Uncertainty in the Current (A), Resistance ( , and Uncertainty in the Resistance ( .
2. Write a formula in Excel to calculate the resistance in terms of the voltage and current.
3. Determine on paper what the formula is to calculate
in terms of the voltage, current, and their
uncertainties.
4. Convert your formula from part (3) into an Excel formula on your spreadsheet.
Check your spreadsheet by calculating the resistance and the error in the resistance, R ±u(R), for the
measured values V = 2.0 ± 0.1 V and I = 100 ± 1 mA.†
What does your spreadsheet say R ± u(R) should be if V = 1.5 Volts, u(V) = 0.1 Volts, I = 60 mA, and u(I)
= 1 mA?
2.4.3.2 Uncertainty in the Resistivity
The formula for resistivity is
.
1. Write a formula on paper for the uncertainty of in terms of , , , and their uncertainties.
2. Add new columns to you Excel spreadsheet with the headings Resistivity (
) and Uncertainty in
Resistivity (
).
3. Implement formulas in Excel to calculate the values for the columns you just created.
NOTE: you will be using your calculated standard deviation
What does your spreadsheet say  ±
.06 ,
650 , and
should be if
20 ?
9.0
for A as the value for u(A).
,
0.1
,
6.00
,
Save the spreadsheet you made and email it to yourself for use in the lab.
2.5
PREPARE FOR THE EXPERIMENT
Read ahead in the lab manual for this experiment, so you can prepare your lab notebook and think about
how to answer the questions in blue.
Include the above uncertainty calculations in the appropriate spots in your lab notebook. Use the
instructions above to set up an Excel workbook, including uncertainty calculations, to use in the
experiment. Email it to yourself.
When you feel ready, test your preparation and your Excel workbook with the Pre-Experiment Quiz on eLearning.
†
The answer should be R ± u(R) = 20 ± 1 ohm. You can also check your spreadsheet by working out the calculation
on paper.
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3
3.1
LABORATORY MANUAL
MATERIALS CHECK OFF LIST
Each small group of (2-3) students will have:
Laptop computer with Excel, LabVIEW, and RStudio
Vernier SensorDAQ
Vernier Voltage Probe
Vernier Current Probe
5 oz container of Play-Doh
4 (four) 2 inch nails
4 (four) 1.5V batteries
1 four-battery holder
2 Alligator clip wires with pigtails
2 wires with banana plugs
4 Alligator clip adapters for banana plugs
flat piece of wood or hard plastic
Ruler
Vernier caliper
Each class will have:
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3.2
Steel wool or sandpaper for cleaning nails
SAFETY AND WASTE DISPOSAL PROTOCOLS
Make sure your lab-bench is dry; there should be no water anywhere on or near your workspace and the
equipment on it. Do not touch bare metal to your skin, and do not mishandle the probes.
Dispose of dried-out Play-Doh in the regular garbage containers. All other materials can be used again.
3.3
EXPERIMENTAL PROCEDURE
In this series of experiments you will first measure the resistivity of Play-Doh, based on the equation for a
cylindrical resistor, and then you will test your predictions of the resistance of unusually shaped resistors.
Discuss the experiment with your partner:
1. Decide whose Excel workbook will be used, and agree on any changes before you begin. Determine
how you will record the estimated error for each measurement and where space for error estimates is
to be included in the Excel workbook.
2. Plan your time. Decide how long to spend on each part of the experiment. The times don’t need to be
exact, but try to keep to your schedule as much as possible.
Q1. Write your planned schedule here:
3.3.1
Check Uncertainties Calculations
Collaborate with your partner to make sure that your spreadsheet will produce the correct uncertainties. If
you are not sure, compare your results with your classmates and check with your instructor to make sure
you all agree on the correct uncertainties for the example values.
3.3.1.1 Uncertainty in the Resistance
The formula for the resistance is
. Refer to the pre-experiment for more details, as needed.
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Q2. What does your spreadsheet say R ± u(R) should be if V = 1.5 V, u(V) = 0.1 V, I = 60 mA, and
u(I)=1 mA?
3.3.1.2 Uncertainty in the Resistivity
The formula for resistivity is
. Refer to the pre-experiment for more details, as needed.
Q3. What does your spreadsheet say  ±
should be if
9.0
6.00 ,
.06 ,
650 , and
20 ?
3.3.2
1.
1.
2.
3.
,
0.1
,
Add a Second Indicator to your VI
Download and open one of your previous VIs that use the SensorDAQ.
Connect the Voltage Probe to channel 1 and the Current Probe to channel 2.
Start LabVIEW and open your VI.
Add a Numeric Indicator for channel 2. Give the Indicator an appropriate title.
3.3.3
Measure How Resistance Varies with the Length of Resistor
3.3.3.1 Build the Circuit
1. Remove the Play–Doh from its container. Begin to work the Play-Doh into a cylinder using your
hands and the hard plastic slab. The shape should be uniform with a diameter approximately 2 cm
and a length approximately 24 cm. It is not necessary to use all the Play-Doh. Use a caliper to
measure the diameter at 5 places along the cylinder. Record each diameter measurement in your
data table.
2. If the nails are rusty or dirty, use the steel wool or sandpaper to clean them. You need a good
connection between each nail and the Play-Doh.
3. Take two of the nails and put them into the ends of the rolled-out Play-Doh so they stick out
horizontally to the left and right.
4. Insert two more nails into the top of the Play-Doh, 14 cm apart. These nails will stick up
vertically. These nails will be connected to the voltage sensor.
5. Set up the circuit indicated in Figure 2. In this case the Play-Doh is the resistor and the voltage
source is the battery pack. For now keep the switch open, but the circuit should be otherwise fully
set up. The horizontal nails will be connected to the battery pack, with the switch.
6. Draw the circuit diagram in your lab notebook. Label the symbols with the physical things they
represent, for instance the voltage source should have the words “battery pack” next to it.
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Figure 1: Simple circuit diagram of a resistor with
a voltage source.
3.3.3.2 Measure Length Dependence
1. Measure and record the distance between the vertical nails. Record the error in your length
measurement.
2. Start the LabVIEW program.
3. Close the switch.
4. Record both the voltage and the current in the appropriate place in your Excel workbook. Include
estimates of their uncertainty.
5. After a few moments stop the LabVIEW program and open the switch so that current does not flow.
6. Remove one of the vertical nails attached to the voltage sensor from the Play-Doh and move it 2 cm
closer to the other vertical nail attached to the voltage sensor.
7. Close the switch, and record the new voltage, current, and distance between the nails.
8. Repeat the previous three steps until you have a total of five voltage/current measurements as a
function of length.
9. Disconnect the battery pack, and remove all of the nails from the Play-Doh.
3.3.4
Testing the Resistivity Equation
3.3.4.1 Length Dependence
Create a heading for this section in your lab notebook.
In this section we will use the data collected in section 3.3.3 where the length of the Play-Doh is changed,
but the cross sectional area is kept fixed.
Recall that equation (1) states the resistance of a cylinder as a function of its dimensions, length L, and
cross-sectional area A and the property of the material, called the resistivity .
(1)
Q4. What type of graph would you expect to see if you plotted R versus L for a fixed, known value of
A? How could you extract the resistivity from this graph?
1. Plot the resistance, R, versus the length, L using either RStudio. You can use your linear fit script.
2. Fit the graph with a straight line and examine the equation.
Q5. Is the intercept equal to 0? Should it be, or can you explain why it isn’t?
Q6. What does the slope represent?
Q7. What is the error in the slope? How did you determine this value?
Q8. What is the significance of the error in the slope?
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3.3.4.2 Direct Calculation of Resistivity and Error Analysis
In the previous section, you calculated the resistivity, and the error in the resistivity, based on a fit line. In
this section you will compare that value, and its error, to the measured resistivity of the Play-Doh, and the
error in that measured resistivity, calculated directly from your data.
Q9. Of all your measurements, which one has the smallest percent error in the length of the Play-Doh?
Explain your reasoning.
Calculate the measured resistivity of Play-Doh using equation (6) and the measurements you identified
above. Use the appropriate error propagation formulas to compute the uncertainty in your measured value
of resistivity.
Q10. Record the results in your lab notebook, and here.
Compare the results from your fit line to the one you gave in answer to the previous question.
Q11. Do the answers agree within the calculated error of your measurements?
In the next lab, you’ll compare these measured values and their uncertainties with one based on area
dependence.
3.4
POST-LAB ASSIGNMENT
In the next lab, you’ll be experimentally determining the effect of cross-sectional area (A) on the
resistance of Play-Doh.
Q12. Given today’s results and the equation for the resistance of a cylinder, make a hypothesis about how
A will be related to R for a fixed length. Explain your prediction.