Lab 8 Permittivity of Free Space and RC Time Constant CPII S24
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8
Permittivity of Free-Space and RC Time Constant
Purpose -
Students will gather data to dtermine the permittivity of Free-Space and the
Time Constnat for an RC-circuit.
Equipment
•
Multi-meter
•
Wire Leads (8)
•
Alligator Clips (6)
•
Resistors in boxes (aprrox.
10K
⌦
&
30K
⌦
)
•
Capacitor (to be measured experimen-
taly)
•
Battery ”D”-Cell
•
Battery Holder
•
Plastic Bracket
•
Plstic Ruler
•
Timer
8.1
Measuring Permittivity of Free Space
One of the most profound experiences a scientist can have is to complete an experiment
that measures one of the fundamental values of the universe.
In this case we will use a
simple parallel plate capacitor to measure the permittivity of free space of an electric field
between two conductors, where
✏
o
is given to be
✏
o
= 8
.
85
x
10
-
12
Nm
2
C
2
.
Relevance between lab and life:
From your study of Biology you are most likely
familiar with the term dielectric constant. In this case, the dielectric constant value is also
the relative permittivity. Relative permittivity is a measure of how much an insulator can
be polarized and is the permittivity value of the material over
✏
o
. For example water,
H
2
O
is 80.
8.1.1
The Theoretical Result
To measure
✏
0
, you will plot the capacitance of the parallel plate capacitor vs. the inverse
of the separation
1
d
between the capacitor plates. This will produce a straight line and the
slope of the line is related to
✏
o
. The capacitance of a parallel plate capacitor is
C
=
✏
o
A
d
where
A
is the area of one of the plates and
d
is the separation between the plates.
Analysis 1: If the capacitance,
C
, is plotted against
1
d
, what is the theoretical slope of the
line? (This will be a symbolic expression.)
slope
=
99
8.1.2
The Measurement
Use the setup at your station to measure the value of capacitance for an air-filled capac-
itor for 6 plate separations as demonstrated by your lab TA at the start of lab. You are
measuring a very small capacitance, so it is important to keep the leads in about the same
configuration since the capacitance of the leads can be an important e
↵
ect. Experiment
with your setup so you can get reproducible results. The meter must be zeroed before the
measurement begins, but should not be re-zeroed during data-taking.
Di
↵
erent plate separations are created by placing spacers at the corners of the plates. The
spacers have a non-trivial dielectric constant, so as little of the spacer should be placed
between the plates as possible. Using the calipers, measure the thickness of a spacer. You
may want to measure several and take the average, record the value below.
thickness
=
WARNING: YOU MUST ZERO THE METER BEFORE
YOU BEGIN THE MEASUREMENT
Data Table 1 - Capacitance and Separation
Spacers
Separation
d
(cm)
Separation
d
(meters)
1
/d
(meters
-
1
)
Capacitance (F)
1
2
3
4
5
6
8.1.3
Drawing a Good Graph
A well drawn graph should
•
Have a title.
•
Have axis labels with units.
•
Have ranges selected so the data mostly fills the graph.
•
Have a best fit line drawn. A best fit line has equal amounts of data on both sides.
8.1.4
Ploting the Data
Using Excel plot a good graph of your data.
Which Goes On Which Axis
The variable YOU control in the experiment, the inde-
pendent variable, should be plotted on the horizontal axis. In this case, you controlled
the separation, so plot 1
/d
on the horizontal axis and
C
, the dependent variable, on
the vertical axis.
100
Don’t Force Zero
A physics experiment does two things; It extracts parameters by fit-
ting a model and it tests the model. The only time it would be appropriate to force
zero is when you have absolute confidence your model completely captures the ex-
perimental setup, that is there are no stray fields or systematic e
↵
ects that you don’t
know about. This is certainly not the case in this experiment.
8.1.5
Analyze the Data
Draw a BEST FIT straight line through the data. Calculate the slope of the line using as
large a rise and run as possible.
Analysis 2: Record the slope of your line from your plot. Report the slope with the correct
units.
slope
=
Measure 1: Measure width,
w
, and length,
l
, of the plate
w
=
l
=
Calculation 7: Calculate the area of the plates:
A
=
Calculate 8: Compute
✏
0
from your slope. Show your work.
✏
o
=
Calculation 9: Compute the error in your measurement,
(
measured
-
theoretical
)
theoretical
x100%
%
Diff
=
101
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8.2
RC Time Constant
Simple resistor and capacitor circuits are used in many of today’s electronics as timing
devices and to maintain voltage and current values in circuits. An example that you use
regularly is windshield wipers. When you adjust the speed (timing) of the wipers, you are
actually changing the resistance in an RC circuit. Another example that you may come
across in the medical field is pacemakers. Artificial pacemakers provide electrical signals
with the correct time constant to regulate a heartbeat. Pacemakers have sensors that can
change the time constant to boost heart rate during exercise.
The time constant,
⌧
, for a simple circuit is defined at the value of the total capacitance
multiplied by the total resistance in the circuit.
⌧
=
RC
(8)
When a capacitor with charge stored on one of the capacitor plates is connected in a closed
loop, the charge stored is allowed to flow through the loop onto the other plate. However,
the current in the system is not constant because the work needed to move the charge
from one plate to the other increases as the ability to do work (voltage) decreases across
the plates.
This means that the voltage across the plates of the capacitor is constantly
changing but is not a linear function.
It is an exponential function related to the time
constant of the circuit.
The voltage across the capacitor is maximum when the closed loop is first connected. Dur-
ing the first time constant, the voltage drops from 100% to 36.6% of the maximum voltage,
a di
↵
erence of 63.4%. During the second time constant, voltage drops from 36.6%to 13.5%
of the maximum. By the end of the third time constant, the voltage has dropped to 5.0%
of the maximum voltage.
The capacitor is considered to be discharged by the 5th time
constant. See the diagram below for a detailed plot of a discharging capacitor.
Voltage as a function of Time Constants - Discharging RC Circuit
102
In this lab experiment, students will discharge a capacitor through a series of resistors large
enough to slow the discharge time so that we can observe the voltage reading across the
capacitor and record the voltage at ten di
↵
erent time intervals. The voltage as a function
of the time will be plotted to display the time constant curve and then linearized in order
to obtain the time constant value from the slope of the plot.
8.2.1
Setting up the Circuit
1. Using the multi-meter and the capacitance meter, measure the values of the compo-
nents to be used in the circuit.
•
capacitor
C
1
=
μ
F
•
resistor
R
1
=
k
⌦
(approx. 10k
⌦
)
•
resistor
R
2
=
k
⌦
(approx. 30k
⌦
).
2. Using the components and wires available, build the series circuit (a loop) in Figure
1.
Figure 1. RC Circuit with Battery connected across Capacitor with open switch.
3. Using the switch (two disconnected wires), the battery, wire leads, and clips, connect
the battery across the capacitor, so that when the switch is closed the capacitor
charges.
4. Connect two wire leads to the voltage meter, a black lead in the ground (COMM)
and a red lead in the Voltage (V). Set the voltage meter to DC and use the 10V
range.
Connect the leads to the capacitor using the clips.
Be sure to connect the
leads to the capacitor so that the voltage reading on the multi-meter is positive.
8.2.2
Collecting Data
1. Close the switch (connect the battery wires) and watch the readout on the multi-
meter. It should jump up to a maximum readout of around 1.5 - 1.6 V dependent on
the strength of the battery. Again be sure to connect the positive end of the battery
to the positive end of the capacitor. Record the maximum reading in Data Table 1.
2. Open the switch and start your timer at the same time. Allow the timer to run until
the voltage reading is about 10% of the maximum reading, around 0.15 - .016 V.
Record the total time in Data Table 1.
103
3. Charge the capacitor again by closing the switch so that the meter reads the maximum
voltage.
This time when you allow the capacitor to discharge you will record the
voltage at equally spaced time intervals. To determine the time intervals, take the
maximum time and divide it by 10 to create 10 equally spaced intervals. (For example,
if the total time is 130 seconds to go from max voltage to 10% of the max voltage,
then your time intervals are 13 seconds long. So the time intervals will be: 0s, 13s,
26s, 39s, etc...) Record these time intervals in the data table. Disconnect the battery
and start the timer, recording the voltage for each time interval.
Data Table 1 - Two resistor time-decay data.
Trial 1 (
R
1
+
R
2
)
Max Voltage (
V
o
)
Resistance Total (
R
1
+
R
2
)
Total Time (s)
Time Interval =
TotalTime
10
(s)
Trial 1
Time
(s)
V oltage
, V(t)
ln
V
(
t
)
V
o
Time
1
- 0 s
V
o
=
Time
2
-
Time
3
-
Time
4
-
Time
5
-
Time
6
-
Time
7
-
Time
8
-
Time
9
-
Time
10
-
4. Repeat the experiment (steps 1 -3) , but this time only use one Resistor, the larger
of the two. Record all data in Data Table 2.
Data Table 2 - One resistor time-decay data.
Trial 2 (
R
1
orR
2
)
Max Voltage (
V
o
)
Resistance Total
Total Time (s)
Time Interval =
TotalTime
10
(s)
104
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Trial 2
Time
(s)
V oltage
, V(t)
ln
V
(
t
)
V
o
Time
1
- 0 s
V
o
=
Time
2
-
Time
3
-
Time
4
-
Time
5
-
Time
6
-
Time
7
-
Time
8
-
Time
9
-
Time
10
-
8.2.3
Data Analysis
1. Using Excel, make a plot (Plot 1) of the Voltage as a function of the Time Intervals
from Data Table 1. Be sure to include all components of a good graph. You should
have your TA sign o
↵
on your graph and then email a copy to yourself for reference.
You will need it for the lab report.
TA Signature (Plot 1):
Analysis 1: Describe the relationship between voltage and time for Plot 1.
2. The voltage as a function of time is not a linear function. In fact the voltage drops o
↵
based on a time constant defined by the total resistance and the capacitance of the
circuit. In this case we are discharging a capacitor so the maximum voltage across
the capacitor occurs at
Time
0
. The equation for this is given by Equation 9.
V
(
t
) =
V
o
e
(
-
t/
⌧
)
(9)
Where V(t) is the voltage at some time t,
V
o
is the maximum voltage and
⌧
is the time
constant for the circuit and is given by,
⌧
=
R
T
⇥
C
. If we want to experimentally
find the value of the capacitor, then we need to make a plot where the slope contains
the time constant. To do this we need to transform the equation for discharging a
capacitor from an exponential form to a linear form using the following steps:
3. Divide both sides by
V
o
.
V
(
t
)
V
o
=
e
(
-
t/
⌧
)
(10)
105
4. Take the natural log function (ln) of both sides.
ln
(
V
(
t
)
V
o
) =
-
t
⌧
(11)
5. Calculate
ln
(
V
(
t
)
V
o
) for all time intervals in Data Table 1.
6. Using Excel, make a plot (Plot 2) of Equation 11, plotting
ln
(
V
(
t
)
V
o
) of the results in
Data Table 1 as a function of Time Intervals (t). Be sure to include all components
of a good graph. You should have your TA sign o
↵
on your graph and then email a
copy to yourself for reference.
TA Signature (Plot 2):
Analysis 2: Describe the relationship between
ln
(
V
(
t
)
V
o
) and the time interval.
Analysis 3: What is the slope of the Plot 2? Look at Equation 11 for help.
Calculate 1: Use the slope and the value of your resistors to calculate the value of
the capacitor. Magnitude of units is important.
Capacitor:
(F)
106
Analysis 4: Compare your experimental value to the measured value.
%Di
↵
erence:
7. Answer the following using the data from Data Table 2.
Analysis 5: from your data in Data Table 2, what do you predict will happen to the
slope for only one resister? (less steep, more steep, stay the same, etc...)
8. Using Excel, make a plot (Plot 3) of Equation 11, plotting
ln
(
V
(
t
)
V
o
) of the results in
Data Table 2 as a function of Time Intervals (t). Be sure to include all components
of a good graph. You should have your TA sign o
↵
on your graph and then email a
copy to yourself for reference.
TA Signature (Plot 3):
Analysis 6: What is the slope of the Plot 3? Look at Equation 11 for help.
Calculate 2: Use the slope from Plot 3 and the value of your resistor to calculate the
value of the capacitor.
Capacitor:
(F)
Analysis 7: Compare your experimental value to the measured value.
%Di
↵
erence:
107
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Question 1: Did your slope match your prediction in Analysis 6? If not, why?
Lab Instructor Signature
Score
/ 10 points
108
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