Resistance and Resistivity Part 1
A playful exploration of the extensive property of resistance and the intensive property of resistivity.
CONTENTS
1
Objectives ............................................................................................................................................. 1
1.1
Prerequisite Skills and Knowledge ............................................................................................... 1
1.2
Research Skills .............................................................................................................................. 1
1.3
Learning Objectives ...................................................................................................................... 2
2
Pre-Experiment ..................................................................................................................................... 2
2.1
Resistivity ..................................................................................................................................... 2
2.2
Measurement with Vernier Calipers ............................................................................................. 2
2.3
Measurement Error and Calculations ............................................................................................ 3
2.3.1
Standard Deviation................................................................................................................ 3
2.3.2
Propagating Uncertainties ..................................................................................................... 4
2.3.3
Propagated Uncertainties in Resistance and Resistivity ....................................................... 5
2.4
3
Prepare Lab Notebook .................................................................................................................. 5
Laboratory Manual................................................................................................................................ 6
3.1
Materials Check Off List............................................................................................................... 6
3.2
Safety and Waste Disposal Protocols............................................................................................ 6
3.3
Experimental Procedure ................................................................................................................ 6
3.3.1
Check Uncertainties Calculations ......................................................................................... 6
3.3.2
Add a Second Indicator to your VI ....................................................................................... 7
3.3.3
Measure the Dependence of Resistance on the Length of Resistor ...................................... 7
3.3.4
Measure the Dependence of Resistance on the Cross-Sectional Area of the Resistor .......... 8
3.3.5
Testing the Resistivity Equation ........................................................................................... 9
3.3.6
Direct Calculation of Resistivity and Error Analysis ............................................................ 9
3.4
Post-Lab Assignment .................................................................................................................. 10
1
1.1
OBJECTIVES
PREREQUISITE SKILLS AND KNOWLEDGE
Students should be competent in algebra.
1.2
RESEARCH SKILLS
After this lab, students will have had practice in:
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1.3
Using a Vernier scale caliper
Measuring current and voltage
Reading a circuit diagram
Propagating error
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 concept 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
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 ρ of a material is defined by its relationship to resistance R,
(1)
𝑅𝑅 = ρ
L
A
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 (Ω m). In this experiment we
will first measure the resistivity of Play-Doh and then examine the effect of shape on resistance.
Based on the formula, which would have a greater resistance, a thick cylinder or a thin cylinder?
2.2
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
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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?
2.3
MEASUREMENT ERROR AND CALCULATIONS
2.3.1
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)
𝐴𝐴̅ =
1
∑ 𝐴𝐴𝑖𝑖
𝑁𝑁
To calculate the error in this value, use the standard deviation σ:
(2)
𝜎𝜎𝐴𝐴 = �
1
∑(𝐴𝐴̅
𝑁𝑁−1
− 𝐴𝐴𝑖𝑖 )2
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2.3.1.1 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.
2.3.2
Propagating Uncertainties
Suppose you wanted to add two measurements: x = 2 m ± 0.2 m and y = 13 m ± 1 m. The sum of the
measurements is clearly 15 m, but what is the uncertainty?
Which measurement do you think has the most impact on the uncertainty in their sum?
To calculate the contribution of each uncertainty in addition or subtraction, use the formula given below.
Always add the squared uncertainties, though, even for subtraction problems.
(3)
𝑢𝑢(𝑥𝑥 + 𝑦𝑦) = �𝑢𝑢(𝑥𝑥)2 + 𝑢𝑢(𝑦𝑦)2
In the above formula the notation 𝑢𝑢(𝑥𝑥) represents the uncertainty in 𝑥𝑥 and 𝑢𝑢(𝑦𝑦) represents the uncertainty
in y.
Plugging in the given values, the uncertainty in the sum is
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𝑢𝑢(𝑥𝑥 + 𝑦𝑦) = �(0.2 𝑚𝑚)2 + (1 𝑚𝑚)2 = �1.04 𝑚𝑚2 = 1.02
For the measurement 2 m ± 0.2 m the uncertainty, ± 0.2 m , is 10% of the value measured. For this
measurement the estimated percent error is 10%. Calculate the percent error in y and in the sum on your
own.
When you combine numbers the uncertainty in the result tends to increase. Thus, the formula for
combining the uncertainties is called an error propagation formula.
There are error propagation formulas for pretty much any arithmetic operation you can think of. Below
are the error propagation formulas for multiplication and division, which will also be used for this lab.
(4)
(5)
2.3.3
Propagated Uncertainties in Resistance and Resistivity
For this experiment, you will also need to calculate the propagated uncertainties in the calculated values
of resistance and resistivity.
2.3.3.1 Uncertainty in the Resistance
𝑉𝑉
𝐼𝐼
The formula for the resistance is 𝑅𝑅 = .
1. Create columns in Excel for the Voltage (V), Uncertainty in the Voltage (V), Current (A),
Uncertainty in the Current (A), Resistance (𝛀𝛀), and Uncertainty in the Resistance (𝛀𝛀).
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 𝑢𝑢(𝑅𝑅) in terms of the voltage, current, and their
uncertainties.
4. Convert your formula from part (3) into an Excel formula on your spreadsheet.
What does your spreadsheet say u(R) should be if V=1.5 Volts, u(V)=0.1 Volts, I=60 mA, and u(I)=1 mA?
2.3.3.2 Uncertainty in the Resistivity
The formula for resistivity is 𝜌𝜌 =
𝑅𝑅𝑅𝑅
.
𝐿𝐿
1. Write a formula on paper for the uncertainty of 𝜌𝜌 in terms of 𝑅𝑅, 𝐴𝐴, 𝐿𝐿, and their uncertainties.
2. Add columns in Excel with the headings Resistivity (𝛀𝛀𝛀𝛀 ) and Uncertainty in Resistivity (𝛀𝛀𝛀𝛀).
3. Implement formulas in Excel to calculate the values for the columns you just created.
NOTE: you will be using your calculated standard deviation 𝜎𝜎𝐴𝐴 for A as the value for u(A).
What does your spreadsheet say 𝑢𝑢(𝜌𝜌) should be if 𝐴𝐴 = 9.0 𝑐𝑐𝑚𝑚2 , 𝑢𝑢(𝐴𝐴) = 0.1 𝑐𝑐𝑚𝑚2 , 𝐿𝐿 = 6.00 𝑐𝑐𝑐𝑐, 𝑢𝑢(𝐿𝐿) =
.06 𝑐𝑐𝑐𝑐, 𝑅𝑅 = 650 𝛺𝛺, and 𝑢𝑢(𝑅𝑅) = 20 𝛺𝛺?
Save the spreadsheet you made and email it to yourself for use in the lab.
2.4
PREPARE LAB NOTEBOOK
Read ahead in the lab manual for this experiment, so you can prepare your lab notebook.
Include the above uncertainty calculations in the appropriate spots in your lab notebook.
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3.1
LABORATORY MANUAL
MATERIALS CHECK OFF LIST
Each small group of (2-3) students will have:
Laptop computer with Excel and LabVIEW
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 experiment 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. You will be using the VI you made in the previous lab period. Download that VI and open it in
LabVIEW on your laptop.
3.3.1
Check Uncertainties Calculations
Collaborate with your partner to make sure that your spreadsheet will produce the correct uncertainties.
3.3.1.1 Uncertainty in the Resistance
𝑉𝑉
𝐼𝐼
The formula for the resistance is 𝑅𝑅 = . Refer to the pre-experiment for more details, as needed.
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Q2. What does your spreadsheet say u(R) should be if V=1.5 Volts, u(V)=0.1 Volts, I=60 mA, and
u(I)=1 mA?
3.3.1.2 Uncertainty in the Resistivity
The formula for resistivity is 𝜌𝜌 =
𝑅𝑅𝑅𝑅
.
𝐿𝐿
Refer to the pre-experiment for more details, as needed.
Q3. What does your spreadsheet say 𝑢𝑢(𝜌𝜌) should be if 𝐴𝐴 = 9.0 𝑐𝑐𝑚𝑚2 , 𝑢𝑢(𝐴𝐴) = 0.1 𝑐𝑐𝑚𝑚2 , 𝐿𝐿 = 6.00 𝑐𝑐𝑐𝑐,
𝑢𝑢(𝐿𝐿) = .06 𝑐𝑐𝑐𝑐, 𝑅𝑅 = 650 𝛺𝛺, and 𝑢𝑢(𝑅𝑅) = 20 𝛺𝛺?
3.3.2
Add a Second Indicator to your VI
1. Connect the Voltage Probe to channel 1 and the Current Probe to channel 2.
2. Start LabVIEW and open your VI.
3. Add a Numeric Indicator for channel 2. Give the Indicator an appropriate title.
3.3.3
Measure the Dependence of Resistance on 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 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 don’t actually plug the batteries into the circuit, but the circuit
should be otherwise fully set up. These nails will be connected to the battery pack.
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. Plug the battery pack into the circuit.
4. After a few moments stop the LabVIEW program and disconnect the battery pack.
5. Record both the voltage and the current in the appropriate place in your Excel workbook. Include
estimates in their uncertainty.
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. Record the new voltage, current, and distance between the nails.
8. Repeat steps 4 and 5 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
Measure the Dependence of Resistance on the Cross-Sectional Area of the Resistor
3.3.4.1 Build a New Resistor to Measure Area Dependence
1. Take the unused portion of the Play-Doh and combine it with the portion used previously. Reshape
this larger lump into a cylinder, first with your hands and then with the wood slab. Roll the Play-Doh
lump down until it has a diameter of about 3.5 cm. Measure this diameter in five places and record
these values in your Excel workbook.
2. Reconnect the nails as you did in the previous section. Place the two vertical, voltage sensor nails 6
cm away from each other.
3. Plug in the battery pack.
4. Record the voltage and current readings, their uncertainties, and the distance between the nails in your
Excel workbook.
5. Remove the battery pack, then remove the vertical nails and roll the play-Doh out until its diameter
has been reduced by about 0.5 cm. Measure the new diameter in five places and record these values
on your data sheet.
6. Reinsert the vertical nails, again 6 cm away from each other. Plug the battery pack back in.
7. Record the voltage and current readings, their uncertainties, and the distance between the nails in your
Excel workbook.
8. Repeat steps 5 through 7 three more times, to get a total of five (5) voltage/current measurements for
different diameters of the Play-Doh.
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3.3.5
Testing the Resistivity Equation
3.3.5.1 Length Dependence
Create a heading for this section in your lab notebook.
In this section we will use the data collected in section 3.3.2 where the length of the Play-Doh is varied,
but the cross sectional area is kept fixed.
You’ve seen the equation for the resistance of a cylinder in terms of its dimensions and the resistivity.
𝐿𝐿
𝐴𝐴
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?
𝑅𝑅 = 𝜌𝜌
1. Plot the resistance, R, versus the length, L.
2. Use this Excel to fit the graph with a trend line.
3. Use the trend line to compute the resistivity. Show your work in your lab notebook.
Q5. Is the intercept equal to 0? Should it be, or can you explain why it isn’t?
3.3.5.2 Area Dependence and Data Transformation
Create a heading for this section in your lab notebook.
For this section, we will work with the data taken in section 3.3.3 which all had roughly the same length.
1. Average the different lengths into a single value and record this in your lab notebook.
The resistance is supposed to be inversely proportional to the cross-sectional area of the cylinder; because
of this, the graph of R versus. A is less simple than that of R versus L. One way to get around this is to
perform a transformation on our data.
2. Define a new variable X to be equal to 1/A.
Q6. How does the resistance depend on X? What shape would we expect a graph of R vs. X to have?
5. For a fixed length, create a new column in Excel for the variable X. Use a formula to calculate X from
your values of A. Label the column heading for X with the appropriate units.
6. Plot the resistance R versus X.
7. Plot a trend line and compute the resistivity.
Q7. Are your two results for the resistivity of Play-Doh consistent with each other? If not, then how big
is the difference?
3.3.6
Direct Calculation of Resistivity and Error Analysis
In the previous section, you calculated the resistivity based on a trend line. This is a valid approach, but
we have not yet taught you how to evaluate the uncertainty in the slope of a trend line. In this section you
will calculate the resistivity of the Play-Doh directly from your data.
Q8. Of all your measurements which one has the smallest percent error in the length of the Play-Doh?
Explain your reasoning.
Q9. Of all your measurements which one has the smallest percent error in the area? Explain your
reasoning.
Calculate the resistivity in terms of both measurements you identified above. Use the error propagation
formulas to compute the uncertainty in the resistivity. Record both results in your lab notebook.
Q10. Which result has a smaller uncertainty for the resistivity? Is this what you expected?
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3.4
POST-LAB ASSIGNMENT
Compare your results with other groups in your class.
Q11. Were the results fairly uniform? If not, suggest possible reasons for the variation in the results.
What could be done to improve accuracy and precision of measurements? What factors cannot be
corrected for?
Decide, as a class, on a value for the resistivity of Play-Doh.
Q12. Record the class value and the standard deviation here, and in your lab notebook.
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