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Department of Physics & AstronomyUndergraduate Labs
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1/2/2012
RC Circuits
When a capacitor is connected to a voltage source, charge will
accumulate on the plates of the capacitor until the voltage
across the capacitor is equal to the voltage across the source.
The
charge
on
the
capacitor
plates
is
proportional
to
the
voltage across the capacitor’s terminals. The constant of
proportionality is the capacitance .
Capacitance is a function of the geometry of the capacitor and the
material properties of the dielectric that is sandwiched between the
plates. For a capacitor with plate area , separation containing a
dielectric between the plates of dielectric constant , the capacitance is
Real capacitors (figure on the right) oftentimes look like plastic cans as
opposed to parallel plates. However, the plastic is simply insulation and the
inside consists of long strips of conducting foil wound around a dielectric. This
configuration enables a large surface area of conductor to be stored in a
small amount of space.
Capacitors are useful for many reasons, but most of all because they can be used to store electrical
energy in the form of a static charge
on the plates. Capacitors can be charged by connecting them to a
voltage source and discharged at a later time. However, capacitors cannot be charged or discharged
instantaneously. The amount of time to charge/discharge depends on the resistance of the circuit.
Below is a diagram of a capacitor being charged by a voltage source. During charging, current will flow
through the resistor as negative charge accumulates on the bottom plate, which repels an equal amount
of charge off the top plate. Charge will accumulate until the voltage across the capacitor is .
/
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Summing voltage drops around the circuit gives
0
This can be rewritten (substituting /) as the differential equation
0
This has the solution
1 / Recognizing that is the maximum charge that can be stored on the capacitor we can write
1
The quantity is the RC‐time constant that dictates how fast the capacitor can be charged. The
figure below shows a few plots of / for various values of the time constant .
Disconnecting the capacitor from the voltage source leaves the capacitor with a charge on the
plates. This charge can be used at a later time to power some electric device, for example. When the
capacitor is discharged, as shown below, a current will flow through the resistor as electrons flow from
the – plate to the + plate. This will continue until the charge on both plates is zero.
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Similar to before, we can sum voltage drops around the circuit:
0
This can be rewritten (substituting /) as the differential equation
This has the solution
/
Again, we see the quantity dictates how fast charge on the capacitor decays. The figure below shows
a few plots of / for various values of the time constant .
In this lab you will observe charge flowing to and from the capacitor in real‐time and observe similar
plots to those shown above.
Goals of this Lab
1. Understand how a capacitor obtains and stores charge
2. Understand how charge flows through a circuit
3. Understand how the capacitance and resistance of a circuit affects charge flow
4. Observe charge flow in real‐time
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Lab Materials
Power Supply
The power supply will act as a battery that can provide a variable
(i.e. user defined) voltage.
Banana Connector Patch Cord
Electrical connections can be made by the sticking the plug of one
into the socket of another.
Alligator Clips
Alligator clips will be used to make electrical contact in your circuit.
Switch
Electrical
connection
to
two
different
circuits
will
be
made
via
a
switch.
Multimeter
The multimeter will primarily be used for troubleshooting your
circuit and measuring resistances.
Differential Voltage Probe
These can be interfaced to LoggerPro for automatic data collection
of voltage differences in a circuit.
LoggerPro
LoggerPro will be used to run the differential voltage probe and used
for data analysis. LoggerPro
Resistors
Resistors will be used to regulate the rate of charging and
discharging a capacitor.
Capacitors
Charge from the power supply will be stored in capacitors.
Light bulbs
Light bulbs will be used to ensure that circuits are properly
connected.
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Overview
You will build the circuit shown below. The circuit consists of two branches and a central capacitor .
The left branch contains a voltage source and a resistor . The right branch contains a resistor
only. The capacitor can be connected to either branch by throwing a switch to the left or right. Closing
the switch
on
the
left
branch
connects
the
capacitor
to
a voltage
source
and
will
charge
the
capacitor.
After the capacitor is charged, the switch can be opened. Charge is now stored on the capacitor. The
capacitor can be discharged by closing the switch on the right branch.
During both charging and discharging you will monitor the charge on the plates by measuring the
voltage across the capacitor. You will connect a voltage probe across the terminals of the capacitor as
shown below
The voltage is related to charge via the relationships . Thus, the voltage will also show an
exponential rise or decay governed by the time constant as the capacitor is charged or discharged,
respectively. We will exploit this to measure the capacitance of several different capacitors by observing
these exponential rises and decays of voltage.
Voltage
To computer
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Lab Procedure
Set up the power supply and LoggerPro
The voltage probes max out at 6V so we want to make sure we apply a lower voltage than this. If you
have a Vernier power supply (small black box), it automatically outputs 5V so go to step 4. If you are
using the PASCO Power Supply follow steps 1‐3:
1. If you are using the PASCO Power Supply set the function generator to DC mode by clicking the
downward arrow button () on Range until the display shows 00000.
2. Set the multimeter to DC voltmeter mode and connect probes to the terminals of the power
supply.
3. Dial up the voltage on the power supply to 5.0V. Leave the dial at this setting!
4. Run LoggerPro and set up the experiment for continuous data collection at 100 samples/s. Go to
Experiment
Data
Collection…
and
check
Continuous
Data
Collection
and
set
the
Sampling
Rate to 100 samples/s.
Build a branched circuit that can charge or discharge a capacitor
1. Choose two resistors and and a capacitor and build the circuit shown below. Measure
the values of and with the multimeter.
Note: You will use only one switch. The diagram shows two to help you visualize how the
physical switches work. When the physical switch is closed, it is equivalent to closing the top and
bottom switches in the circuit diagram below.
Note: Some of the capacitors we are using are electrolytic capacitors that use an electrolyte
(instead of metal) for one plate. These capacitors have a polarity, meaning that one side must be
connected to the positive terminal of the power supply. This terminal is marked by a +.
2. Connect the voltage probe as shown and observe the voltage displayed in LoggerPro. The
voltage across the capacitor should be zero (with small fluctuations about zero). Discharge the
capacitor by shorting out the terminals with wire if necessary.
Voltage
To computer
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Prelab Exercise 1:
You are given the following set of capacitors and resistors. Which combination will cause the capacitor
to charge at the fastest rate when you close the left‐hand switch? What resistance should you use if
you want a capacitor to charge at the slowest rate? What resistor should you use if you want a
capacitor to discharge at the fastest rate?
Capacitance
Resistance
130×106 µF 500
150 µF 1 M
1500 pF 10 k
Verify that your circuit works as expected
1. Replace and with light bulbs.
2. Throw the switch to the left. You should observe a flash of light through the bulb as your
capacitor is charged. Depending on how large
is, the flash may last for several seconds.
3. Close the switch to the right. You should observe a flash of light with identical (provided that the
light bulbs are about the same) duration.
4. If you observed the flashes, your circuit is properly connected. Replace the light bulbs with
and .
Prelab Exercise 2:
Why should you observe only a flash of light, as opposed to a continuous glow, when the switch is
thrown to the left? After all, the light bulb is directly connected to the power supply.
Charge the
capacitor
1. Start collecting data with .
2. Close the switch to the left to charge the capacitor. Observe the voltage vs. time. Autoscale the
graph with as needed.
Note: Depending on your values of and , the capacitor could charge very fast or very slow.
If nothing happens, something in your circuit isn’t connected properly.
3. Wait until your capacitor is fully charged (i.e. until the voltage is constant). This typically takes
about 5 time constants (5. Don’t stop collecting data!
Prelab Exercise 3:
How many time constants (give an integer number) will you have to wait before the voltage across the
capacitor is within 99% of its initial value?
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Discharge the capacitor
1. Throw the switch to the right to discharge the capacitor. Observe the voltage vs. time.
Note: Again, depending on your values of and , the capacitor could discharge very fast or
very slow. If nothing happens, something in your circuit isn’t connected properly.
2. Wait until your capacitor is fully discharged (i.e. until the voltage is about zero). Again, this takes
about 5 time constants (5). Stop data collection ( ) when it is fully discharged.
Analyze your results
You should observe two curves showing the charging and discharging of the capacitor. They should look
similar to the curves in the introduction.
1. Fit a Natural Exponential ( ) to the voltage rise curve (charging). Use the brackets [ ] on
either side of the data to select the region for the fit. You may have to rescale the horizontal axis
to zoom
in
on
the
curve.
Do
this
by
left
clicking
the
first
and
last
tick
label
of
the
horizontal
axis
and changing the value. Don’t include the flat region to the left of the curve in the fit!
2. Fit a Natural Exponential ( ) to the voltage decay curve (discharging). Use the brackets [ ] on
either side of the data to select the region for the fit.
3. Determine the capacitance of your capacitor from each of your exponential fits.
4. Create a column that calculates the charge by going to Data New Calculated
Column and entering in the appropriate values in the Equation box. Use the average value
you obtained from the previous step, not what is marked on the side of the capacitor! You can
use
values
from
columns
that
contain
data
(i.e.
the
potential
column)
by
clicking
.
5. Repeat this experiment using a different combination of and values. If you didn’t do so,
choose values that will fully charge/discharge the capacitor over the timescale of minutes.
Questions
1. Attach plots of the charging/discharging curves including fits. Label the resistances for each curve.
2. Make a table that lists the resistances and and your measured value of capacitance . 3. Do your capacitance values match for charging and discharging? If not, what could be causing the
discrepancy? Do your capacitance values match with what is labeled on the side of the capacitor?
4. What was the maximum charge
stored on each capacitor? How many electrons does this
correspond to?
The
charge
of
an
electron
is
1.6 10C.
Feedback
Please leave feedback about this lab. What did you like best? What did you like least? Did anything go
wrong? What did you feel like you learned in this lab?
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