Download The Capacitor

The Capacitor©98
Experiment 3
Objective: To study the properties of capacitance as they concern the discharge of a
capacitor through a resistor.
DISCUSSION:
A capacitor is a device for storing charge. In its simplest form it may be thought
of as two parallel conducting plates which lie very close to one another but which are
electrically insulated from one another. With the use of an "electron pump", electrons are
drawn from one side of the plate (which then becomes deficient in electrons and is left
positively charged) and deposited on the other (which then has a surplus of electrons and
is thus negatively charged).
C
R
e-
direction of electrons
i
C
"Electron Pump"
Voltage Source
+
Figure 2: Discharging a capacitor
Figure 1: Charging a capacitor
A schematic drawing of the charging of a capacitor is shown in Fig. 1. The plates
are equally though oppositely charged. If the electron pump is removed and the circuit
left open, the charges remain separated on the two plates. However if the circuit is
closed, the electrons on the negatively charged plate rush back to the other plate to join
again and neutralize the positive charges left there. When there is a resistance in this
circuit, as shown in Fig. 2, the flow of electrons is impeded, and the neutralization of the
plates does not occur instantly.
The ability of a capacitor to hold charge is called capacitance. It is found that the
quantity of charge Q which is stored on the plates is proportional to the voltage V of the
electron pump. The constant of proportionality is defined to be the capacitance C. That
is,
(1)
Q  CV
The unit of capacitance is the farad and is equal to a coulomb/volt.
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(In practice, the size of the plates of a capacitor which could hold a coulomb of
charge under the action of an electron pump of one volt would be extremely large. For
example, if the plates were square and separated by a vacuum of 1 millimeter, the length
of their sides would be over 6.5 miles! The capacitance of commercially produced
capacitors are in the neighborhood of 10-6 farads (microfarads or f) to 10-12 farads
(picofarads or pf). )
It can be shown that if several capacitors are connected in series, the effective
capacitance, CTotal , of the combination is given by
1
1

CTotal

C1
1
C2

1

C3
(2)
Here, C , C , C , etc. are the capacitances of the individual capacitors. On the other
1
2
3
hand, if the several capacitors are connected in parallel, the effective capacitance of the
combination is given by
(3)
C
 C  C  C 
Total
1
2
3
When a capacitor discharges through a resistor, as in Fig. 2, it acts like an electron
pump whose voltage gets progressively smaller because the charge held on its plates
becomes progressively smaller. This pumping voltage V is given at any instant by
V 
q
i
q
(4)
C
where q is the charge on the plates at the given instant. Since q steadily decreases, so
also does V. The current i that is set up in the resistor R at any given instant is given by
Ohm's law. Thus
V
i
(5)
R
Hence,
(6)
RC
Since the current measures the flow of charge from the capacitor and since q
represents the charge remaining on the capacitor, it is clear that as q becomes smaller, so
also does the rate at which it leaves the capacitor. The flow of charge from the capacitor
occurs at an ever decreasing rate.
If the current in the resistor at the moment the circuit in Fig. 2 is first closed is
called Io, then the current i at some later time t is given by the expression
i  I o e t RC
(7)
where e is the base of the natural logarithms (e = 2. 718282…). Eq. (7) expresses the
exponential decay of the current. The half-life of the circuit is the time in which the
current i has dropped to half its initial value Io. A graph of current versus time is shown
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Taking the natural logarithms of both sides of
Eq. (7) yields
 i 
(8)
t   RC ln  
I
 o
Letting i=Io/2, we have that = 0.69315RC.
Hence, determining the half-life of the current
enables us to find RC.
The circuit used in this experiment is
shown in Fig. 4. When the switch S is closed, the
voltage V across the capacitor is whatever the
voltage is across the tapped-off portion r of the
rheostat, which itself is across an external source
of constant voltage. The voltage V is also the
voltage maintained across the resistance R. The
steady current measured by the galvanometer
while the switch is closed is I = V/R.
When the switch is opened, the external
source no longer maintains the charge on the
capacitor, and the capacitor discharges through
the resistor. The galvanometer at any instant now
measures the discharging current i.
Discharging a Capacitor
1.2
i=Io
t=0
1
Current (amps)
in Fig. 3. The current never quite reaches zero
in theory because the charge on the capacitor
never quite becomes zero.
0.8
i=Io /2
t=0.693RC
i=Io /4
t=1.386RC
0.6
0.4
0.2
0
0
20 Time (s) 40
60
Figure 3: Sample graph of Current vs Time
for discharging capacitor.
+
_
Power
Amplifier
r
V
C
+
switch
_
R
G
Figure 4: Circuit for measuring the
discharging current of a capacitor.
EXPERMENTAL SETUP:
1. Setup the Signal Interface as shown in Fig. 5.
a. Plug the power brick into the back of the Interface, then into the strip on your
table.
b. Connect the white cable into the back of the Interface then into your
computer’s USB port.
c. Plug a voltage sensor into Analog Channel A.
d. Turn on the Interface (switch is on the back).
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Figure 5: PASCO Interface setup.
2. Start DataStudio. It should be located somewhere under your start menu.
3. When prompted, choose ‘Create Experiment’.
4. Set up the software to recognize the hardware:
a. In the software, add a “Voltage Sensor” to Channel A.
b. Notice that there is a list of Data options on the upper left (usually). Also notice
that there is a list of Display options on the bottom left (usually).
c. Click and drag the “Graph” icon to “Voltage, ChA (V)”.
d. Move and resize the graph window so you can see everything.
5. Setup the Sampling Options:
a. Find the ‘Set Sampling Options” function. In version 1.9.7r8, this can be found by
clicking on the “Sampling Options…” tab above the interface image.
b. Under ‘Delay Start’, click on Data Measurement.
c. From the drop down menu immediately below ‘Data Measurement’, choose
‘Voltage, ChA (V)’.
d. From the next drop down menu, choose ‘Fall below’.
e. Set the value to 9.7V.
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EXERSICES FOR CAPACITOR:
1. Build a charging circuit for the capacitor (refer to Fig. 6) using a 200f capacitor and
a 100k resistor. On one side, you will hook up the DC power supply so you can
charge your capacitor to roughly 10V. On the other side, you will have a resistor and
capacitor (RC) circuit to discharge the capacitor over a period of time. Note that the
electrolytic capacitors are polarized and must be connected properly to avoid an explosion!
2. Throw the switch to the power supply side. This will charge the capacitor to the
potential set on the supply (10V).
3. Click the ‘Start’ button to begin recording data.
V
4. Throw the switch to the resistor side. DataStudio
will record the voltage across the capacitor as it
discharges through the resistor.
5. Click on the ‘Stop’ button when the voltage has
reached 0.1V.
_
+
+
_
6. At this point, you should have a graph on your
screen of the voltage across the capacitor as a
function of time.
a. Notice the value of the initial voltage V .
0
b. Calculate V 2 .
0
c. Magnify the area of the graph where the
voltage crosses V 2 , until you can read the
0
Power
+ Amplifier _
Figure 6. A charging circuit for the
capacitor using a DPDT switch.
value of the half-life off the graph.
d. Compare this measurement to the theoretical calculation = 0.69315RC.
7. Repeat the exercise using other resistors with the 200 f capacitor.
8. Repeat the exercise using other resistors with the 100f capacitor.
9. Wire the two capacitors in series and repeat the exercise for the combination.
a. What is the effective capacitance of the combination?
b. Calculate the percent error between your measured value and the theoretical value
from Equation 2.
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10. Wire the two capacitors in parallel and again repeat the exercise for the combination.
a. What is the effective capacitance of the combination?
b. Calculate the percent error between your measured value and the theoretical value
from Equation 3.
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