August 7, 2007
CAPACITANCE
PURPOSE: There are three objectives in this experiment:
ƒ to determine the precise values of some commercially available
capacitors and hence to check the formulae for combining capacitors
in parallel and in series.
ƒ to investigate how the capacitance of a parallel plate air-dielectric
capacitor varies with plate separation.
ƒ to measure the dielectric constants of several insulating materials.
REFERENCES:
Introduction to Electrodynamics, 3rd Edition, D. J. Griffiths:
p 103 to 106; 183 to 186
Electricity and Magnetism, 3rd Edition, W. J. Duffin: p 108 to 120
University Physics, 11th Edition, Young & Freedman:
P 909 to 918; 922 to 928
Handbook of Chemistry and Physics, 78th Edition, CRC Press Inc.
APPARATUS:
3 commercial capacitors
multimeter
3 terminal boards
6 leads
parallel plate capacitor
16 paper spacers
steel weight
calipers/micrometer
2 pieces of acrylic
2 pieces of Styrofoam
6 pieces of Bristol board
INTRODUCTION:
Capacitors are commonly used in a variety of electric circuits: they are used to tune
radio receivers, as filters in power supplies, to eliminate sparking in automobile ignition
systems, and as energy storage devices in electronic flash units. In this experiment we
will investigate these devices in more detail.
When an uncharged finite conductor removed from other objects is given a charge Q, its
potential is raised from zero to a certain value V (with respect to zero potential at
infinity). The value Q/V is called the capacitance of the conductor, C:
C≡
Q
V
CAPACITANCE-1
[1]
August 7, 2007
For example, a spherical charged conductor with charge Q has an electric potential
outside the sphere given by:
V=
Q
4πε o r
[2]
At the surface, the potential is given by equation [2], substituting r = R, the radius of the
sphere. By the definition of capacitance, and a reference position at infinity, the
capacitance of an isolated charged sphere is therefore:
C=
Q
Q
=
= 4πε o R
V Q
4πε o R
[3]
Note that the capacitance of an isolated charged sphere is proportional only to its radius,
and is independent of the charge on the sphere. This is true in general: capacitance is an
inherent property of the conductor, and depends only on the geometry and the material
with which the device is made, with some dependence on environmental factors such as
temperature, pressure, humidity, etc..
Typically capacitance is discussed in reference to two conductors carrying charges of
equal magnitude and opposite sign. In this scenario, a potential difference, ∆V, exists
between the two conductors due to the presence of this charge separation. Equation [1]
then becomes:
C≡
Q
∆V
[4]
where Q is the magnitude of the charge on each conductor. Capacitance is a measure of
the amount of charge the conductors will hold for a given potential difference, and by
definition, capacitance is a positive quantity. The SI unit for capacitance is the
Coulomb/Volt or farad, symbol F. In practice, the submultiples µF and pF tend to be
more convenient.
In this experiment, we will be investigating parallel plate capacitors. These devices
consist of two parallel metal plates of equal area A, separated a distance d as shown in
figure 1. One plate carries a charge of +Q, the other carries a charge of –Q. Therefore,
each plate has a surface charge density of:
σ=
Q
A
[5]
Assuming that the plates are very close together, in comparison with their length and
CAPACITANCE-2
August 7, 2007
width, it can be shown that the electric field between the plates is:
r σ
E =
[6]
εo
With a uniform electric field, the potential difference between the two plates is then:
r
σd
∆V = E ⋅ d =
εo
[7]
and by combining equations [7], [5], and [4], the capacitance of the parallel plate device
is given by:
C=
εo A
[8]
d
Again, the capacitance is only dependent on geometric factors: the area of the plates (A)
and the distance between them (d).
-Q
+Q
Area = A
d
Figure 1: Schematic depiction of a parallel plate capacitor.
In the preceding discussion, it was assumed that the space between the two conductors
was evacuated. This is not always the case; in many devices, a non-conducting material
called a dielectric is inserted between the two conductors to increase the capacitance. If
the dielectric completely fills the space between the conductors, the capacitance
increases by a dimensionless factor κ, which is called the dielectric constant of the
material:
C = κ Co
[9]
CAPACITANCE-3
August 7, 2007
where Co denotes the capacitance in the absence of the dielectric.
In particular, for a parallel plate capacitor with a dielectric, the capacitance is given by:
C =κ
εo A
d
[10]
Table 1 summarizes some values of the dielectric constant. Note that these materials all
have dielectric constants greater than unity: this implies that the presence of these
dielectrics will always serve to increase the inherent capacitance of the devices to which
they are added.
Table 1: Dielectric constants of various substances (20°C)
Vacuum
Air
Acrylic
Bakelite
Neoprene
Nylon
Paper
Bristol board
Plastic
Polyethylene
Polystyrene
Pyrex glass
Silicone oil
Styrofoam
Teflon
Transformer oil
Water, distilled
1.00000
1.00059
2.7 – 4.5
4.9
4.0 – 6.7
3.4
1.9 - 3.7 depending on grade
1.8 – 2.0 depending on air content
2.6 - 3.6 depending on type
2.25 - 2.3
2 – 2.8
5.6
2.5
1.03
2.1
22
80
In many applications, more than one capacitor is used in a circuit. It is often therefore
useful to be able to calculate the equivalent capacitance between two points such as A
and B in the circuit, where the equivalent capacitance is again defined by equation [4].
However, we can now express ∆V as VA – VB, the potential difference resulting from the
transference of Q from B to A. In other words, the equivalent capacitance is the
capacitance of a single capacitor which, if connected between A and B, would produce
the same effect as the original configuration. Determining equivalent capacitance
involves two principles: conservation of charge and the path-independence of potential
difference. In this experiment we will be investigating two special cases: capacitors in
series and capacitors in parallel.
CAPACITANCE-4
August 7, 2007
Capacitors in series
Figure 2 shows a set of capacitors in series, with the capacitors represented by their
circuit diagram symbol: two thick parallel lines of equal length.
A
B
C1
C2
C3
Figure 2: three capacitors in series.
When point A is connected to the positive terminal of a battery, and B is connected to
the negative terminal of the battery, charge begins to flow. Electrons from C1 flow to
the positive terminal, leaving a net +Q on the left-hand plate of C1, while electrons from
the battery flow to C3, leaving a net –Q on the right-hand plate of C3. Conservation of
charge ensures that the left-hand plate of C1 and the right-hand plate of C3 accumulate
charge of equal magnitudes. As negative charge builds up on the right-hand plate of C3,
an equivalent amount of negative charge is forced off the left-hand plate of C3, leaving it
positively charged. The negative charge leaving the left-hand plate of C3 accumulates
on the right-hand plate of C2, and so on. The overall effect is that all the capacitors in
series have ±Q on their conducting surfaces, as shown in figure 3.
A
+Q
-Q
+Q
C1
-Q
+Q
C2
+
-Q
B
C3
–
∆V
Figure 3: Charge distribution among capacitors in series connected across a
battery.
The potential differences between the two conductors of each capacitor is then:
∆V1 =
Q
C1
∆V2 =
Q
C2
and
∆V3 =
Q
C3
[11]
By adding ∆V1, ∆V2 and ∆V3, we get the total potential difference between A and B or:
CAPACITANCE-5
August 7, 2007
Q Q Q
∆V = V A − VB =
+
+
C1 C 2 C3
[12]
and by definition, this implies that the equivalent capacitance between A and B is:
V − VB
1
1
1
1
= A
=
+
+
Ceq
Q
C1 C 2 C3
[13]
In general, for n capacitors in series:
1
=
Ceq
n
∑C
1
i =1
[14]
i
Capacitors in parallel
Figure 4 shows a set of capacitors in a parallel configuration. The left hand plate of each
capacitor is connected to the positive terminal of a battery, and are therefore all at the
same electric potential. The right hand plates are all connected to the negative terminal,
and are also all at the same potential. Therefore, the potential difference across each
capacitor is the same when connected in parallel.
C1
A
B
C2
C3
+
–
∆V
Figure 4: capacitors arranged in a parallel configuration.
As with the discussion of capacitors in series, when these devices are connected to the
terminals of the battery, charge begins to flow. The net result is that the left hand plates
become positively charged, while the right hand plates become negatively charged. The
flow of charge ends when the potential difference across each capacitor is exactly the
amount across the terminals of the battery, ∆V. Rearranging equation [4], this implies
that the maximum charge of each capacitor, achieved when the potential difference is
∆V, is given by:
Q1 = C1 ⋅ ∆V
Q2 = C 2 ⋅ ∆V
and
CAPACITANCE-6
Q3 = C3 ⋅ ∆V
[15]
August 7, 2007
The total charge transferred in reaching this steady state is:
Q = Q1 + Q2 + Q3
[16]
due to conservation of charge. Therefore, recognizing again that ∆V = VA – VB, the
equivalent capacitance between A and B is:
Ceq =
C ⋅ ∆V + C 2 ⋅ ∆V + C3 ⋅ ∆V
Q
= C1 + C 2 + C3
= 1
V A − VB
∆V
[17]
In general, for n capacitors in parallel:
n
Ceq =
∑C
i
[18]
i =1
Note:
Capacitor value codes (in pF):
3rd digit Multiplier
0
1
1
10
2
100
3
1000
4
10,000
5
100,000
6, 7
not used
8
0.01
9
0.1
Letter
D
F
G
H
J
K
M
P
Z
Tolerance
0.5 pF
1%
2%
3%
5%
10%
20%
+100, -0%
+80, -20%
Example: If a capacitor is marked 105, this means 10 * 100,000 = 1 × 106 pF = 1000 nF
= 1 µF. The letter added to the value is the tolerance as indicated in the table above.
EXPERIMENT
NOTE: after each measurement of capacitance, turn the meter off before disconnecting
the device you have measured.
Part 1: measuring commercial capacitors
1-1.
Connect the 1st commercial capacitor to the multimeter, and turn the dial to the
capacitance setting:
CAPACITANCE-7
August 7, 2007
Record the capacitance measured, as well as the uncertainty.
1-2.
1-3.
1-4.
1-5.
1-6.
Repeat 1-1 with the 2nd and 3rd capacitors.
With the leads disconnected from any commercial capacitor, measure the inherent
capacitance of the meter. Record this value and its uncertainty in your notebook.
This capacitance is in parallel with the commercial capacitors when measured, so
correct your measured values accordingly.
Connect the three capacitors in series, and record the overall capacitance
including uncertainty. Remember to correct for the meter’s capacitance.
Connect the three capacitors in parallel, and record the overall capacitance
including uncertainty. Remember to correct for the meter’s capacitance.
Based on the corrected values of the individual capacitances measured
previously, calculate the expected values for combining the devices in series and
in parallel, including uncertainties. Record these calculations in your notebook.
You should have a table that looks something like:
Configuratio
n
C1
C2
Measured
capacitance
(corrected)
Uncertainty
Calculated
capacitance
C3
Uncertaint
y
C1, C2, C3 in
series
C1, C2, C3 in
parallel
Part 2: Investigating the relationship between capacitance and plate spacing
Now you will use the meter to determine the capacitance of the parallel plate capacitor
for a number of different plate separations ranging from d ≈ 0.5mm to d ≈ 6.0mm. Make
sure that the paper spacers do not cover the metal plate, otherwise there will be an error
in the capacitance measured value.
2-1. Starting with two spacers, measure the total spacer thickness, d, with the caliper or
the micrometer. Record this value in a table in your lab book, along with the
associated uncertainty.
2-2. Position a pair of spacers on either side on the bottom plate. Make sure that the
spacers do not overlap onto the plate itself. Place the top plate face down on top,
CAPACITANCE-8
August 7, 2007
taking care to ensure that the plates are aligned.
2-3. Weight the top plate after placing the spacers.
2-4. Record the capacitance and the associated uncertainty in your notebook.
2-5. Add one additional spacer on each side of the lower plate, and repeat steps 2-1
through 2-4. Continue this procedure until you have measured capacitance with
all spacers available.
Plot the results as a linear graph, 1/d vs. C, plotting 1/d on the y axis as it has the
larger uncertainties. From this graph, determine εo. Compare this value with the
tabulated εo. (Note: the area of the plates is 0.0103 m2 ± 0.7%.)
Part 3: Investigating the effect of using a dielectric material
Again, use the meter to investigate the capacitance with various dielectrics inserted. The
procedure is similar to that in part 2, except that for simplicity you will only collect
capacitance values at two plate separations for each material.
3-1. Measure the thickness of one piece of a given dielectric. Record this value and its
uncertainty in your lab book. (NOTE: for testing Bristol board, use a minimum of 3
sheets for your first plate separation measurement.)
3-2. Place the dielectric on the lower plate, and position the upper plate on top, taking
care to ensure that the plates are aligned. The dielectric should cover the entire
lower plate. Weight the top plate down.
3-3. Measure the capacitance of this configuration, recording the value and its
uncertainty in your lab book. NOTE: if the capacitance meter reading does not
stabilize fairly quickly, disassemble the capacitor and try again.
3-4. Now measure the thickness of two pieces of this dielectric. Repeat the capacitance
measurement with two pieces, and record the necessary data. (For Bristol board,
your second plate separation measurement should involve more than 3 sheets.)
3-5. Repeat steps 3-1 to 3-4 for the remaining dielectrics.
For each material, use the two thickness measurements to determine the
dielectric constant by comparison with the data collected previously with the air
capacitor.
Thus determine the dielectric constant of each material, including the
uncertainty.
CAPACITANCE-9
August 7, 2007
Informal lab report expectations for the Capacitance lab
Maximum length for report = 5 pages on 8.5”x11” paper, stapled together, min. 12 pt font size.
Follow the organizational structure below.
Part 1 Measuring commercial capacitors
•
•
give a table of measured and calculated capacitances, with their uncertainties, for both
individual capacitors and series and parallel network of capacitors; a skeleton of such a
table is shown in the lab write-up to show how it could be organized
show calculations of the net capacitance in series and parallel, and the associated
uncertainties.
Part 2 Investigating the relationship between capacitance and plate spacing
•
•
•
give a table of corrected measured capacitances together with the associated plate
spacing: give the uncertainties here as well.
provide a plot of inverse plate spacing versus the capacitance data, together with a linear
fit of the data: ensure the plot has a title, proper axis labels with units, and that the linear
fit slope and intercept are quoted on the graph, together with their uncertainties
o the linear fit must be calculated using weighted linear regression, whether it is
done by hand in a spreadsheet, or by a software program (note that Excel does
not perform weighted linear regression in its straight line fits)
o if the linear fit is done by hand, show the spreadsheet calculations
o if the linear fit is done by a software package, state what the package is, and
what parameters were used, if any
show the calculation of εo and its uncertainty from the slope of the fit and its
uncertainty; compare to the accepted value.
Part 3 Investigating the effect of using a dielectric material
•
•
•
•
give a table of corrected measured capacitances and their corresponding plate
separations, with uncertainties, for each dielectric material
give a mathematical derivation relating the dielectric constant of a given dielectric
material to that of air using the capacitance equations for parallel plate capacitors, one
with air, the other with a dielectric material; use this derivation to calculate the dielectric
constant for each of the materials tested
give a sample calculation of the dielectric constant for one of the dielectric materials
give a table summarizing the dielectric constants of the materials, together with their
uncertainties; compare to accepted values
Part 4 Conclusions
•
•
•
•
summarize results: how well do measured and calculated capacitances agree in part 1?
How close is your estimate of εo to the accepted value in part 2?
Are the dielectric constants you calculated in part 3 reasonable?
Discuss systematic errors.
CAPACITANCE-10