Capacitors
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 units of
Farads (F), (1 Farad = 1 Coulomb/Volt).
If a capacitor of capacitance C (in farads), initially charged to a potential V (volts) is connected
0
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 changed.
Figure 1
As current flows, the charge q on the capacitor plates is depleted, reducing the potential
difference across the capacitor, which in turn reduces the current. This process creates an
exponentially decreasing voltage, modeled by
V (t) = V0 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 its final value exponentially,
modeled by
V (t) = V0 (1 ! e
!
t
RC
)
(3)
The same time constant RC describes both the rate of charging and discharging.
OBJECTIVES
_
Measure an experimental time constant of a resistor-capacitor circuit.
_
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
Logger Pro software
2-Low leakage capacitors (1 to 10 µF)
2-kΩ resistors (50 to 150 kΩ)
Vernier Voltage Probe
Power supply
Connecting wires
Single-pole, double-throw switch
Computer
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 Figure 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.
4. Prepare the computer for data collection by opening “Exp 27” from the Physics with
Computers experiment files of Logger Pro 2.1. The graph displayed will have a vertical axis
corresponding to potential (scaled 0 to 4 V) while the horizontal axis corresponds to time
(scaled 0 to 10 s).
5. Charge the capacitor for 30 s or so with the switch in the position as illustrated in Figure 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 3 volts.
Wait until the potential is constant.
6. Click
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 an exponentially decreasing function.
7. To compare your data to the model, select only the data after the potential has started to
decrease by dragging a rectangle across the graph; that is, omit the initial constant portion.
Click the curve fit tool
, and from the function selection box, choose the Natural
Exponential function, A*exp (–C*x ) + B. Click “Try Fit” and inspect the fit. Click “Done”
to return to the main graph window.
8. 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 equation to the
mathematical model for a capacitor discharge proposed in the introduction,
V (t) = V0 e
!
(2)
t
RC
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.
9. Print the graph of potential vs. time.
10. The capacitor is now discharged. To monitor the charging process, click
. As
soon as data collection begins, throw the switch the other way. Allow the data collection to
run to completion.
11. This time you will compare your data to the mathematical model for a capacitor charging,
V (t) = V0 (1 ! e
!
t
RC
)
(3)
Select the data beginning after the potential has started to increase by dragging a rectangle
across the graph. Click the curve fit tool
, and from the function selection box, choose
the Inverse Exponential function, A*(1 - exp (–C*x )) + B. Click “Try Fit” and inspect the
fit. Click “Done” to return to the main graph window.
12. Record the value of the fit parameters in your data table. Compare the fit equation to the
mathematical model for a charging capacitor.
13. Now you will repeat the experiment with a different resistor and capacitor.
14. Repeat the experiment with two capacitors hooked in series. Capacitors in series add as the
reciprocal. That is,
1 &1# & 1 #
=$ !+$ !
C $% C1 !" $% C2 !"
(4)
15. Repeat the experiment with two capacitors hooked in parallel. Capacitors in parallel add
directly. That is,
C=C +C
1
2
(5)
DATA TABLE
R1 = ______________
R2 = ______________
C1 = ______________
C2 = ______________
Fit Parameters
Trial
A
B
C
1
C
Resistor
Capacitor
Time
Constant
Percent
Error
R (Ω)
C (F)
RC (s)
%
Resistor
Capacitor
Time
Constant
Percent
Error
R (Ω)
C (F)
RC (s)
%
Discharge 1
Charge 1
Discharge 2
Charge 2
Capacitors in Series
Equivalent Capacitor = _______________ (Equation 4)
Fit Parameters
Trial
Discharge 1
Charge 1
A
B
C
1
C
Capacitors in Parallel
Equivalent Capacitor = ______________ (Equation 5)
Fit Parameters
Trial
A
B
C
1
C
Resistor
Capacitor
Time
Constant
Percent
Error
R (Ω)
C (F)
RC (s)
%
Discharge 1
Charge 1
QUESTIONS
1. A capacitor is a device that can store energy as it stores charge. The amount of energy stored
is just equal to work done in moving the charge into 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
1
) Q V = ( ) C V2.
2
2
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?
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.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
-
3. (a) Find the equivalent capacitance of the group of capacitors shown in the figure below.
(b) Find the charge stored on the 8 µF capacitor.
24 V
6µF
8µF
2µF
2µF
4µF