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A capacitor is a device for storing charge and energy. It consists of two conductors insulated from each other. A typical capacitor is called a parallel-plate capacitor and is symbolized -| |- or -|(- . It consists of two conduction plates separated by a small distance. When the two plates are connected to the two wires of a charging device, e.g.. a battery, positive charge is transferred to the plate connected to the positive terminal of the charging device and an equal magnitude of negative charge is transferred to the other plate. This flow of charge continues until the potential difference between the conductors equals the applied potential difference, e.g. the potential difference between the battery terminals. The amount of charge ( Q ) depends on the geometry of the capacitor and is directly proportional to the potential difference ( V ). The proportionality constant is called the capacitance ( C ).
Q = C V
The gap between plates is usually filled with a dielectric material to increase the capacitance. Charge is measured in coulombs and voltage in volts, so capacitance has units of coulombs/volt that is called a farad or F.
This experiment will investigate the discharge of a capacitor through a resistance.
The increase or decrease of charge on the capacitor is exponential in character (as a function of time in this case). This is an important type of behavior in nature since it occurs whenever the instantaneous change in a quantity is only dependent on the amount present at that time. Population and bacterial growth, money in a savings account, and energy use may follow such a law.
Written in electrical symbols the charge, as a function of time, on a discharging capacitor is given by:
Q = Q
0 e !
t /( RC ) 6.1 where
Q is the instantaneous charge at time t ,
Q o
is the initial charge at the beginning of the time interval t ,
R is the resistance in the circuit,
C is the capacitance of the capacitor used, and e is the base of the nature logarithm, e = 2.71828… .
The product RC is called the time constant of the circuit. It has units of ohms-farads,
(which also equals seconds), and is the time required for the charge on the capacitor Q to fall from Q o
to Q o
/ e , as can be seen by substituting time, t = RC into Equation 6.1 above . The value 1/ e ~ 0.37 , so at t = RC , Q = 0.37 Q o
. Since this contains an awkward, rounded constant value, t
½
or half time is usually determined experimentally instead of RC time directly. t
½
is defined as the time necessary for the
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Q to increase or decrease to ½ of its maximum value. When the charge has decayed to ½ of the original charge, we have Q
= 1
2
Q
0
which can be solved to give
Q
Q
0
=
1
2
.
Replacing Q with the right side of Equation 6.1, we have:
1
2
= t 1 e '
RC
2
= exp $
$
%
&
' t
1
2
RC
!
!
"
# where exp( n ) stands for e to the power n . Taking the ln of both sides (the inverse operation as exp( n ) ) to “cancel out” the exp( n ) operation, we have: ln 1
2
=
!
t
1
2
RC
Rearranging this yields: t
1
2
= 0 .
69 RC 6.2
Two cases will be studied in this experiment. In the first, the exponential discharge or decay is slow enough that measurements can be made with a clock or watch. In the second part, the decay is so fast that an oscilloscope must be used.
Figure 6.1: Circuit used to measure slow discharge of a capacitor. R
1
C = 1 .0
µ F, R v
= 10M Ω .
= 10k Ω ,
The circuit shown in Figure 6.1. will be used to measure slow discharge of a capacitor. The voltmeter here is connected in series with the other components. This is
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an unusual use of a voltmeter (voltmeter are normally placed in parallel to other components). Figure 6.1 shows the voltmeter in terms of its internal resistance. There are obviously many more components inside the voltmeter to allow it to function as a voltmeter, but these are not shown here so we can focus on the effect the voltmeter has on the flow of current through the circuit. Momentarily touching the positive wire from the power supply to the indicated node charges the capacitor. When the power supply is then disconnected, the capacitor will discharge through the resistor and voltmeter. (Note that it is important that the power supply wire be physically disconnected after charging in order for this experiment to function properly.) The size of the current, however, is extremely small and would require a very sensitive meter to record. However, the large resistance of a multimeter used in the voltmeter mode results in a measurable voltage drop with even a small current flow. Using the voltmeter in series thus allows for a reasonable measure of the current change, and therefore the charge change, with time, since the current is proportional to the voltage according to Ohm's Law.
Using a stopwatch and working in teams, readings from the voltmeter should be taken every 2 seconds after the capacitor is charged. (One person watches the time and calls out intervals, one reads the voltmeter out loud; and the third records the voltage.)
Graph the voltage vs. time. We now have three ways to determine the time constant:
1.
From the circuit components R = R
1
+ R v
and C to give RC .
2.
By picking the half time off our graph and using Equation 6.2 to calculate RC .
3.
By using the Excel trendline (display the equation) with an exponential fit.
Compare the RC values (the time constants) from these three methods using the uncertainties in each. Which method is most accurate? Also replot the data using a logarithmic axis for voltage with trendline. This should produce a straight line plot.
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Figure 6.2: Rapid Decay Equipment Setup
When RC times are in the low millisecond range or less, a stopwatch is too slow to be used. To measure such decay times an oscilloscope is usually used. This is an electron device which allows one to monitor the voltages versus time of very rapidly varying signals, such as those inside televisions, radios, computers, and other electron devices.
Instead of an oscilloscope, we will be using Data Studio in the “scope” mode with voltage sensors. The RC circuit is shown in Figure 6.2 . The signal generator is set to produce a square wave output as shown in Figure 6.3(a), where the period T is the time for one complete cycle of this signal. In a sense, the signal generator acts like an intermittent version of the power supply used in the earlier part of this lab: at times it outputs 5 volts, during which it tries to charge the capacitor through the resistor, and at other times it outputs zero volts, behaving as a simple wire and trying to discharge the capacitor. You will need to adjust the period such that the “on” time of the signal generator is sufficient to almost fully charge the capacitor. A good approximate setting is for a half period (the length of the “on” time) equals four time constants of the resistorcapacitor combination: T/ 2 = 4 RC .
Since the signal generator is not calibrated in time periods, but instead in frequency f which is the reciprocal of period ( f = 1/ T ), we see that we need to set the frequency so that T /2 = 1/(2 f ) = 4 RC or f = 1/(8 RC ) .
When the frequency is properly adjusted, the voltage across the capacitor should have a time dependence similar to that shown in Figure 6.3(b).
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Procedure for Rapid Exponential Decay
1.
Wire up the circuit shown in Figure 6.2 with R = 10 k Ω and C = 0.047 µ F and both voltage probes plugged into the computer interface box (sockets A and
B). Make sure the interface box is turned on.
2.
Calculate the RC time constant for this RC combination:
R C = _ ______________ . Also calculate the required signal generator frequency as specified in the equation just above f = ___________________ .
3.
Adjust the signal generator to approximately this frequency, then start up Data
Studio, and on the screen drag both voltage probes into the picture of the interface box (sockets A and B). Then drag the resulting Channel 1 and
Channel 2 data sets into the “Scope” display. A realtime display of the voltages detected by the two voltage sensors versus time should appear on the screen. If not, you may have to click on the “start” button. To make the display steady, click on the second tool bar icon which controls the trigger.
You may need to drag the arrow tip that appears on the left side of the graph to a level that is somewhere between the minimium and maximium voltage values. This arrow tip controls the trigger point, at which voltage the scope starts its graph. You should also experiment with changing the horizontal and vertical scales of the graph by clicking on the double arrow symbol just below the horizontal axis and another set just to the right of the graph. When the graph resembles 6.3(a) and 6.3 (b) superimposed, click on “print” in the “file” menu to print out a copy of your screen.
4.
On the printed graph, mark the point at which the voltage on the exponentially decaying signal has reached ½ its starting (or maximum) value. Clearly label the time interval on the horizontal axis that elapsed during which the voltage dropped from the maximum value to one half this value. Record this time interval: t
½
= ________________ .
Estimate an uncertainty for this measurement.
5.
Calculate the expected half time based on the values of RC t ½ = 0.69 RC = ___________________ .
Estimate an uncertainty for this calculation.
6.
Do the results of steps 4 and 5 above agree with each other within experimental uncertainties?
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Figure 6.3: Step Voltage of Signal Generator (a) and Waveform Response from
Circuit (b)
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