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C APACITANCE
References
Crummet and Western: Physics: Models and Applications, Sec. 25.1-2.
Young and Freedman, Sears and Zemansky: University Physics (10th ed.), Sec. 25.2-3 and 27.5.
Introduction
We wish to determine the capacitance of each of a set of three capacitors. This will be
accomplished utilizing two distinct experimental approaches along with mathematical expressions that
relate the capacitance of a capacitor to various quantities like applied voltage, circuit resistance, and
discharge time. In addition, we will experimentally verify the expressions that relate the overall
capacitance of a given combination of capacitors to individual capacitance values of the capacitors
that are used in the combination.
In the first part, use will be made of the relationship defining capacitance C = Q/V. In particular,
each capacitor will be charged in turn with a Constant Current power supply. This type of power
supply will ensure that a known amount of charge (Q) is transferred onto the capacitor in a given
amount of time. The voltage will be measured as a function of time. A graph of Voltage vs time
should be a straight line, with the slope related to capacitance.
In the second method, a Constant Voltage power supply will be used to charge the capacitor.
After the capacitor has been charged to a desired value, it will be allowed to discharge through a
resistor of known value. The voltage across the capacitor will be monitored as a function of time.
Voltage is expected to behave according to the following formula:
V (t )  Vo e

t
(1)
RC
A graph of this voltage as a function of time would not be a straight line, and as such, would be
difficult to use for a direct calculation of the capacitance. However, it is straightforward to perform
calculations on the data such that the new relationship IS a straight line. This is accomplished by
taking the natural log (ln) of both sides of the equation. This step leads to the following equation.
ln(V (t ))  ln(Vo ) 
t
RC
(2)
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A graph of ln(V) vs. time should therefore produce a straight line with a slope equal to the value –
1/RC. Given that R is easily determined, the slope enables us to calculate C, the capacitance.
Equipment
Constant Current Source (~1 mA)
Constant Voltage Source (~3 V)
Three capacitors (electrolytic)
Resistor (~1000 ohms, verify the value!)
Computer with data acquisition software (LabPro)
Digital multimeter
Double-throw switch
Procedure
CAUTIONS:
 Be sure that the positive side (RED) of the power supply is connected to the red capacitor
terminal. These capacitors have definite polarity.
 Do not charge any capacitor for longer than 10 seconds on the first trial.
 The voltage should not exceed 5.0 V on any trial.
 Remember to discharge the capacitor(s) before every trial. Do this by connecting a wire
directly across the two terminals of the capacitor.
Data Acquisition Setup
1. Plug in the AC adapter power cord to LabPro. A series of 6 beeps and different colored lights
in rapid succession would indicate successful hardware initialization.
2. Connect LabPro to the laptop via the data port type of preference. Your choices are USB and
Serial Port. Both options are on the right hand side of LabPro, with a sliding door to force a
single choice.
3. Attach the voltage probe to the interface. The voltage probe is analog and so should be
plugged into channel one.
4. Start the LoggerPro 2.1 software. If LoggerPro is already installed, then the computer and
program should have already identified the device attached. If the device is not identified, a
prompt to “scan” for the interface (COM 1 – 4, or USB) will appear. Choose scan, and if the
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scan is successful, the line below the input box will change from “Interface: Select Port to
Scan” to “Interface: LabPro.”
If there are any difficulties up to this point, ask for assistance.
Two windows should appear within the Program frame: a Table Window, and a Graph Window.
The right side of the program frame is the Table Window, and it already has two columns lined up in
spreadsheet format. There is a label at the top of the table (Latest Run), and each column is labeled
Potential (V) and Time (s).
As a quick check on the equipment, add a new Window by choosing the Window drop-down
menu, and choosing New Wide Window, and selecting the meter option. Choose the size of the
numbers and the font to be 12 point and 6 point, respectively. This will put a large new window into
view, with small numbers and letters on it. This window gives a continuous update of the sensors
attached. The readings should be Time and Potential, and the reading in Potential should be
fluctuating slightly about a mean value of zero. Touching the leads of the voltage sensor together
should cause the reading to become more steady and closer to zero. Re-size this window to make it
smaller, and less obtrusive, and increase the size(s) of the Graph and/or Table window if needed. If
there is any problem with the readings on the meter at this point, ask for assistance. As a
second check on the equipment, attach the probes directly to the 3 Volt power supply and push the
red button. The reading on the meter should (of course) be close to 3.0 V.
Constant Current Source
1) Connect the current source to the capacitor labeled C1 with the voltage probe leads also
connected to capacitor C1, in parallel with the power supply. Also ensure that there is an
ammeter connected in series with the C1 (see Fig. 1). This allows for monitoring the constant
current source.
2) Set the parameters for the experiment so that a trial data set is taken for 10 seconds. Look at
the data obtained, and choose an appropriate data rate in step 3).
3) Based on the results obtained in step 2), adjust the parameters so that data will be collected
up to the point that the capacitor has a final voltage of approximately 3.0 V.
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RC Circuits
A
Capacitor
Current
Source
Fig. 1
To voltage probe
Capacitor and Constant Current Source
4) Fit a straight line to the data (e.g. see fig. 2) and obtain an estimate of the value of the
capacitance for C1, with the associated uncertainty by using the value of the slope. Make sure
you only fit the straight rising part of the line (hint: Q = i x t = C x V for a constant current
source).
Figure 2
Capacitor charging for constant current power supply
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(Note: Since we are not saving these files, it is very important that you record the capacitance values
obtained from these runs in your lab book).
5) Repeat steps 1) - 4) using capacitor C2.
6) Repeat steps 1) - 4) for the Series Combination of C1 and C2. Remember to discharge EACH
capacitor before a new run.
Constant Voltage Source
1) Connect the capacitor C2, the nominal 1000  resistor (measure and record the actual
resistance), the Constant Voltage power supply, and the voltage probes in a circuit as shown.
Switch
Voltage
Source
Capacitor
Resistor
Fig. 3
To voltage probe
Circuit for RC Discharge Experiment
2) Charge the capacitor to approximately 3.0 V (for a “long” time). You can monitor this voltage
with the meter window in Logger Pro.
3) Start collecting data the same time when you close the switch to discharge the capacitor
through the resistor. Observe the shape of this graph (do not save this file). Look at the data
obtained, and choose an appropriate data rate in step 4).
4) Repeat the experiment with parameters chosen so that the voltage on the capacitor drops to a
value of 1.5 V or lower during the time data is taken. Save this data set and print out a graph
of this data. Also, if your instructor requires, save this data file in an “Excel friendly” format.
5) Repeat step 4 for C1. You do not need to save this data set in an Excel friendly format.
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Open Switch Decay Rate (Optional)
Charge the capacitor C1 to a point that it is at approximately 3.0 V. Open the switch (do NOT
connect the resistor), and monitor the voltage on the capacitor for 400 seconds (the data rate can be
relatively low). Determine the rate at which the voltage on the capacitor decreases with time. Use the
results of this calculation, along with the value of the capacitance obtained previously in the
experiment to determine the rate at which charge leaks off of the plates. This leakage occurs
because the gap between the plates of a capacitor does not have an infinite resistance, but instead an
effective “leakage resistance,” Rint. Record whatever values you will need in order to determine Rint.
Suppose that the actual current used to charge the capacitor in part 1 of the lab were the
measured current - leakage current. How (if at all) would this affect the value of capacitance
calculated?
Anal ysi s
Part I
(If time permits, do this part (steps 1-4) for both capacitors before leaving the lab. You must at least
do the analysis for one capacitor right here in the lab).
1) Arrange the Table Window so that you can calculate ln(V) for the last data set taken for the
discharge setup.
2) Plot that new data (ln(V) vs. t) and fit a straight line to this data. Record the value of the slope
and the uncertainty in the slope.
3) Print these graphs. Make sure, your graphs clearly show the fit, and the fitting parameters
along with their uncertainties (see Appendix).
4) Calculate the capacitance for C1 using the value obtained for the slope. You have to
propagate the uncertainty in the slope to get an uncertainty for the capacitance values.
5) Repeat steps 1) - 4) for capacitor C2.
Part II
1) What are the differences in the behavior of the current supplied by the two types of power
supply used in this experiment (Constant Current, Constant Voltage)? Explain why the
charging graph for the constant current source appears to be a straight line. Justify your
answer with the help of expressions that relate charge to current and to voltage and
capacitance.
2) Compare the capacitance values obtained from the Constant Current and the Constant
Voltage portions. Is the difference in capacitance values obtained from the Constant Current
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and the Constant Voltage portions of the lab large enough to require an explanation? If so,
can you think of a plausible reason for the difference?
3) For the constant current source part of the experiment, are the capacitance values obtained
for the series combination consistent (within error) with values obtained earlier for C1 and C2,
and with the expression that predict values for series combination.
Ceq series
 1
1
 

 C1 C 2



1
Appendix
Checklist
Before leaving the lab, you should
 Have capacitance values noted in your lab book for C1, C2, and their series combination from
the constant current source part of the experiment.
 Have 2 data files saved for C1 and C2 for the RC discharge part of the experiment.
 Have a data file saved in Excel friendly format.
Fits to Subsets of Data
If you want to fit only a portion of the data displayed, then drag the mouse pointer over the region
of data you wish to have fit. (Note you have to get the data inside the box). Once “boxed”, choose
the appropriate fitting option from the Analyze drop-down menu. The procedure is then the same as
discussed above for fitting the full data set.
Equation Calculation / New Columns
Often, we will want to compare data with expected (theoretical) values. In the second part of
today’s lab, we expect to find an exponential relationship between the potential and time (the RC
circuit). LoggerPro could fit an equation to that data, and give a best-fit value to the initial voltage and
to the value of R*C. However, no error bound would be provided with that fit. So, we will make our
own column to store the values of the natural log of the potential. If a graph is then made of ln(V) vs.
t, we expect to see a straight line given by Eq. 2. Make the Table Window active, and choose New
Column, and then Formula. This will add a blank column to the Table Window, once information has
been provided for two tabs. In the first tab choose a name for the new column (not too long), and a
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short name for it (ln(Data)), as well as units (ln(V)). A default color and symbol for use in the Graph
Window will be selected and can be used without changing them.
The relation to be used to calculate the values to appear in the new column will need to be
entered in the Definition tab. Begin by looking at the Functions drop down dialog box, and scroll down
the options to ln(). When chosen, that expression appears in the box labeled Equation. Place the
cursor in between the parenthesis. Now, from the drop-down menu labeled Variables at the bottom of
the screen, choose Potential. Next choose Try New Column, and a nice example of the new graph
will appear in the window. Go back to the options tab and choose ok. A graph of the new column(s)
appears on the screen. Notice that the new column will actually be a new column for each set of data
that you have in the Table Window. The graph will not be auto-scaled, so you should use that
tool right away. The original data can be added back onto the graph if you like by using the View,
Options menu. The ln(V) data can now be fit with the linear function, with the slope and intercept
determined with error bounds. Try that at this time, and write down the equations and standard
deviations obtained from the fits. If the Latest run is displayed on the graph, then the raw data, and
the values calculated using the new column will show up on the graph, as the data is taken. Try this
at this time.