Discharge of a
Capacitor
THEORY
The charge Q on a capacitor’s plate is proportional to the potential difference V across
the capacitor. We express this with
Q=CV
(1)
where C is a proportionality constant known as the capacitance. C is measured in the unit
of the Farad( F), (1 F = 1 coulomb/volt).
If a capacitor of capacitance C (in Farads), initially charged to a potential V (volts) is
connected across a resistor R (in ohms), a time-dependent current will flow according to
Ohm’s law. This situation is shown by the RC (resistor-capacitor) circuit below when the
switch is closed.
0
R
Red
C
Red
+
Voltage
Probe
Vo
Black
Black
Fig. 1
As the current flows, the charge q is depleted, reducing the potential across the capacitor,
which in turn reduces the current. This process creates an exponentially decreasing
current, modeled by
V = Vo e-(t/RC)
(2)
The rate of the decrease is determined by the product RC, known as the time constant of
the circuit. A large time constant means that the capacitor will discharge slowly.
When the capacitor is charged, the potential across it approaches the final value
exponentially, modeled by
V = Vo (1- e-(t/RC))
(3)
The same time constant RC describes the rate of charging as well as the rate of
discharging.
OBJECTIVES
_
Measure an experimental time constant of a resistor-capacitor circuit.
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Compare the time constant to the value predicted from the component values of the
resistance and capacitance.
_ Measure the potential across a capacitor as a function of time as it discharges and as
it charges.
_ Fit an exponential function to the data. One of the fit parameters corresponds to an
experimental time constant.
_
MATERIALS
Computer
Logger Pro
Voltage Probe
connecting wires
2-Low leakage capacitor (1 to 10 µF)
2-kΩ resistors (50 to 150 kΩ)
Power supply
single-pole, double-throw switch
PROCEDURE
1. Measure the values of the capacitors and resistors and record the values in the data
table. The lab instructor will show you how to measure the values.
2. Connect the circuit as shown in Fig. 1 above with the first capacitor and the first
resistor.
3. Connect the Voltage Probe to the Logger Pro and across the capacitor, with the red
(positive lead) to the side of the capacitor connected to the resistor. Connect the black
lead to the other side of the capacitor. It is important that the black terminal from the
logger pro is connected to the black terminal of the power supply.
3. Prepare the computer for data collection by opening “Exp 27” from the Physics with
Computers experiment files of Logger Pro 2.1. A graph will be displayed. The
vertical axis of the graph has potential scaled from 0 to 4 V. The horizontal axis has
time scaled from 0 to 10 s.
4. Charge the capacitor for 30 s or so with the switch in the position as illustrated in Fig.
1. You can watch the voltage reading at the bottom of the screen to see if the potential
is still increasing. Adjust the voltage from the power supply so that the voltage lies
close to three volts. Wait until the potential is constant.
5. Click collect to begin data collection. As soon as graphing starts, throw the switch to
its other position to discharge the capacitor. Your data should show a constant value
initially, then decreasing function.
6. To compare your data to the model, select only the data after the potential has started
to decrease by dragging across the graph; that is, omit the constant portion. Click the
curve fit tool “f(x)=”, and from the function selection box, choose the Natural
Exponential function, A*exp(–C*x ) + B. Click “try fit”, and inspect the fit. If the
curve is a good fist, then click “ok” to return to the main graph window. However, if
black line doesn’t match the red line, try click and hold the brackets and move them
around to improve the curve fit. (The black curve is the curve fit and the red curve is
the data.)
7. Record the value of the fit parameters in your data table. Notice that the C used in the
curve fit is not the same as the C used to stand for capacitance. Compare the fit
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equation to the mathematical model for a capacitor discharge proposed in the
introduction,
V = Vo e-(t/RC)
(2)
Notice that the curve fit C is equal to one over the product of resistor and the
capacitor C. Find the percent error between 1/C (curve fit) and RC.
8. Print the graph of potential vs. time.
9. The capacitor is now discharged. To monitor the charging process, click “collect”.
As soon as data collection begins, throw the switch the other way. Allow the data
collection to run to completion.
10. This time you will compare your data to the mathematical model for a capacitor
charging,
V = Vo (1- e-(t/RC))
(3)
Select the data beginning after the potential has started to increase by selecting the
region of the graph to curve fit by dragging mouse across the graph and highlighting
the section. Click on the “f (x) =” button and selecting the inverse exponent function,
A*(1 – exp (–C*x)) + B. Click “try fit” and inspect the fit. Click “ok” to return to the
main graph window.
11. Record the value of the fit parameters in your data table. Compare the fit equation to
the mathematical model for a charging capacitor.
12. Now you will repeat the experiment with a different resistor and capacitor.
13. Repeat the experiment with two capacitors hooked in series. Capacitors in series add
as the reciprocal. That is,
(4)
1 1
1
= +
C C1 C2
14. Repeat the experiment with two capacitors hooked in parallel. Capacitors in parallel
add directly. That is,
€
C = C1 +C2
(5)
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DATA TABLE
R1 = ______________
R2 = ______________
C1 = ______________
C2 = ______________
Curve Fitting Parameters
Trials
A
B
C
1/C
Resistor
Capacitor
Time
Constant
R(Ω)
C (F)
RC (s)
Resistor
Capacitor
Time
Constant
R(Ω)
C (F)
RC (s)
Percent Error
Discharge 1
Charge 1
Discharge 2
Charge 2
Capacitors in Series
Equivalent Capacitor = _______________ Eqn. 4
Curve Fitting Parameters
Trials
A
B
C
1/C
Discharge 1
Charge 1
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Percent Error
Capacitors in Parallel
Equivalent Capacitor = ______________ Eqn. 5
Curve Fitting Parameters
Trials
A
B
C
1/C
Resistor
Capacitor
Time
Constant
R(Ω)
C (F)
RC (s)
Discharge 1
Charge 1
QUESTIONS
1. A capacitor is a device that can store energy when it stores charge. The amount of
energy stored is just equal to work done in moving the charge onto the capacitor. The
work done increases as more charge is added to the plate. Therefore, the energy stored
on the capacitor is just W= (1/2) Q V = (1/2) C V2. Using the data from your
experiment, calculate the energy stored in the capacitor for the first trial. What became
of that energy when the capacitor is discharged?
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Percent Error
2. Below shows a diagram of a charged parallel plate capacitor without a dielectric
between the plates and the other diagram shows the same parallel plate capacitor that has
a dielectric placed between the plates. (a) Sketch the electric field lines between the
plates on each diagram. (b) If the charge on the plates remains constant while the
dielectric is slipped between the plates, will the voltage on the capacitor with the
dielectric increase, decrease, or remain constant when compared to the capacitor without
the dielectric.
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