Physics 260 Calculus Physics II: E&M
Grist
RC Circuits
Object
In this experiment you will study the exponential charging and discharging of a capacitor
through a resistor. As a by-product you will confirm the formulas for equivalent
capacitance of capacitors connected in series and parallel.
Equipment
Low voltage power supply
Two capacitors (nominally between 0.5 µF and 1 µF)
Resistance Box
Two connection boxes to support the capacitors
Digital multimeter
Stopwatch
Switch
Patch cords
Theory
Consider a capacitor of capacitance C with an initial charge Q0 allowed to discharge
through a resistor of resistance R.
t=0
R
C
Q0
Figure 1
The voltage across the plates of the capacitor decays exponentially with time according to
V(t) = V0 e −t / RC
(Eq. 1)
where V0 is the initial voltage. (The charge on the plates and current to the plates varies in
a similar exponential fashion.) Figure 2 shows a graph of V vs. t for this exponential
decay.
Physics 260 Calculus Physics II: E&M
Grist
V
V0
V0/e
t
τ
(Time constant)
Figure 2
The time constant for the circuit, τ, is the time for the voltage (or current, or charge) to
decay to 1/e (≈ 0.368) of its initial value. It is a measure of the response time for the
circuit.
It is easy to show (see questions) that
τ = RC
(Eq. 2)
It is instructive to take the logarithm of both sides of Eq. 1 to obtain
lnV = lnV0 −
t
RC
(Eq. 3)
Hence a graph of ln V vs. t will yield a straight line with slope equal to –1/RC = –1/τ, as
illustrated in Figure 3.
ln V
ln V0
slope = Š1/RC
t
Figure 3
Physics 260 Calculus Physics II: E&M
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Procedure I: Using the multimeter to measure the resistance and
capacitance of the circuit
1.
I will explain how to use the multimeter to measure resistance and capacitance at the
beginning of the lab. Pay attention! If you blow a fuse, you pay for it.
2.
Set your resistance box to a nominal resistance of 10 MΩ. (Make sure all the other
keys are down; sometimes one will sneak up on you.) Use the multimeter to measure
the resistance more accurately and record in Table 1. The internal resistance of the
voltmeter is very nearly 10 MΩ and you do not need to measure it.
3.
Use the multimeter to measure the capacitance of each of the two capacitors that
were supplied to you.
Procedure II: Observing the exponential decay
1.
Set up the circuit shown below. (The figure shows approximate nominal values; use
the circuit components supplied to you and the measured values from Procedure
I.)The internal resistance of the voltmeter is 10 MΩ so the total resistance of the
right loop is about 20 MΩ. Have the instructor check your circuit before you turn on
the power.
R box - 10 Mž
S
Rint Å 10 k½
Low Voltage
Power Supply
C - 1.0 µF
V
R meter - 10 Mž
6.0 V
Figure 4
Note the unusual circumstance that the voltmeter is not connected in parallel with
any circuit element. This is usually considered a sacrilege, but there is a good reason
for it here. We want as much resistance as possible in the circuit so that you will be
able to measure the exponential decay with a stopwatch. It is also possible to show
(see Questions) that although the reading of the voltmeter in this circuit is not equal
to the voltage across the capacitor it is proportional to it, and that will be sufficient
for our purposes.
Physics 260 Calculus Physics II: E&M
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2.
Close switch S. The internal resistance of the power supply is only about 10 kΩ, so
the voltmeter should read half the power supply voltage almost immediately. (See
Questions.)
3.
Open switch S and observe the voltmeter reading begin to decrease — rapidly at
first, and then more slowly. This time the capacitor is discharging through about 20
MΩ and has a much larger time constant. The response of the circuit is slow enough
that you can time the decay with a stopwatch. Record the voltage at 3.0-s intervals
for a total of about 30 s. Record in Data Table 2.
4.
Repeat steps 2 – 3 twice more and calculate the average voltage for each time.
Record in Data Table 2.
5.
Plot a graph of V vs. t to see the exponential decay explicitly.
6.
Plot a second graph, this time of ln V vs. t, and do a linear regression analysis. What
is the time constant for this circuit, and how does it compare to what you would
expect from the measured circuit values? Remember, the total resistance is now
R = Rbox + Rmeter.
Procedure III: Verify formula for equivalent capacitance of series
combination.
1.
Replace the 1.0-µF capacitor with a series combination of the 1.0-µF capacitor and
the 0.68-µF capacitor. (The figure shows approximate nominal values; use the
circuit components supplied to you and the measured values from Procedure I.)
S
Rint Å 10 k½
Low Voltage
Power Supply
6.0 V
R box - 10 Mž
C 1 - 1.0 µF
V
R meter - 10 Mž
C 2 - o.68 µF
Figure 5
2.
Close the switch so the capacitors charge up almost immediately.
3.
Open the switch and begin timing. Measure the voltage decay every 3.0 s for about
30 s. Record in Data Table 3.
Physics 260 Calculus Physics II: E&M
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4.
Repeat steps 2 – 3 twice more and calculate the average voltage for each time.
Record in Data Table 3.
5.
Plot a graph of ln V vs. t (you do not need a graph of V vs. t) and do a linear
regression analysis. What is the time constant for this circuit, and how does it
compare to what you would expect from the measured circuit values?
Procedure IV: Verify formula for equivalent capacitance of
parallel combination.
1.
Replace the series combination with a parallel combination. (The figure shows
approximate nominal values; use the circuit components supplied to you and the
measured values from Procedure I.)
S
Rint Å 10 k½
R box - 10 Mž
Low Voltage
Power Supply
V
R meter - 10 Mž
6.0 V
C 1 - 1.0 µF
C 2 - o.68 µF
2.
Close the switch so the capacitors charge up almost immediately.
3.
Open the switch and begin timing. Measure the voltage decay every 3.0 s for about
30 s. Record in Data Table 4.
4.
Repeat steps 2 – 3 twice more and calculate the average voltage for each time.
Record in Data Table 4.
5.
Plot a graph of ln V vs. t (you do not need a graph of V vs. t) and do a linear
regression analysis. What is the time constant for this circuit, and how does it
compare to what you would expect from the measured circuit values?
Physics 260 Calculus Physics II: E&M
Grist
Name_______________
RC Circuits
Data Table 1
Circuit component
Nominal Value
Measured Value
MΩ
Rbox
10
Rmeter
MΩ
MΩ
10
MΩ
C1
µF
µF
C2
µF
µF
Data Table 2: Single capacitor
Time (s)
V1 (V)
V2 (V)
V3 (V)
Vavg (V)
0.0
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
30.0
Time constant from graph
Time constant from circuit values
Percent Difference
Attach graphs of V vs. t and of ln V vs. t.
ln Vavg
Physics 260 Calculus Physics II: E&M
Grist
Data Table 3: Series capacitors
Time (s)
V1 (V)
V2 (V)
V3 (V)
Vavg (V)
ln Vavg
Vavg (V)
ln Vavg
0.0
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
30.0
Time constant from graph
Time constant from circuit values
Percent Difference
Attach graph of ln V vs. t.
Data Table 4: Parallel capacitors
Time (s)
V1 (V)
V2 (V)
V3 (V)
0.0
3.0
6.0
9.0
12.0
15.0
18.0
21.0
24.0
27.0
30.0
Time constant from graph
Time constant from circuit values
Percent Difference
Attach graph of ln V vs. t.
Physics 260 Calculus Physics II: E&M
Questions
1. Show that for an exponential decay like Eq. 1 the time for the voltage to decay to
V0/e is given by the time constant shown in Eq 2.
2.
When the switch is closed in Fig. 4 and the capacitor is charging, what is the
equivalent resistance of the circuit? What is the time constant for the circuit? (Do
you see why the capacitor charged almost immediately?)
3.
The voltmeter reads the voltage across its terminals, and in Fig. 4 that is not the same
as the voltage across the capacitor. Show, however, that the voltmeter reading is
proportional to the capacitor voltage. What is the constant of proportionality?
3.
Each time you closed the switch and charged the capacitor (or capacitor
combination) the voltmeter read only about half the power supply voltage. Explain
this in view of your answer to Question 2.
Grist