TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
2. PARALLEL PLATE CAPACITOR
1. Objective
Determining capacitance of a capacitor and dielectric constant of the insulating material.
2. Equipment needed
Experiment stand with reference capacitor C e and capacitor with unknown capacitance,
dielectrical plates, electrostatic voltmeter or standard multimeter, calliper or micrometer.
3. Theory
In case the space between capacitors plates is filled with homogeneous dielectric material, its
capacitance can be calculated, as follows:
C=
εε 0 S
(1)
,
d
where S is plates surface, d – distance between plates, ε – dielectric constant of the dielectric
F
(plate) and ε 0 – permittivity constant in SI-system units (ε 0 = 8,85 ⋅ 10 −12 ) .
m
Figure 2.1
Formula (1) is not applicable when the space between capacitor's plates is not filled uniformly
with a homogeneous dielectric material (like air). Let us consider a case when a dielectric
plate with thickness d’ is placed between and in parallel to the plates of a capacitor with air
dielectric (fig. 2.1). Capacitance of a capacitor can be calculated as follows:
C=
C0
,
d'  1 
1 +  − 1
d ε

(2)
where C 0 is capacitance of the capacitor without a dielectric plate.
Let us derive dielectric constant of the dielectric plate from formula (2):
ε=
1
d C
1 − 1 − 0
d' 
C



.
(3)
It follows from (3) that capacitances C 0 and C are needed to determine dielectric constant of
the insulating material ε .
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
Capacitance of unknown capacitor in this experiment is determined as follows. Reference
capacitor C e with known capacitance and charged to some voltage is connected parallel to
the unknown capacitor C X . Measuring change of potential on capacitors (connected in
parallel) is enough to calculate capacitance of capacitor C X . Connection diagram of
experiment is shown on fig.2.2.
Closing contacts of switch K1 charges reference capacitor C e up to the voltage U 0 of a DC
supply. Voltage between reference capacitor C e plates is measured with voltmeter V with
very big internal resistance. Every voltmeter has a (parasitic) capacitance CV (shown with
dashed line on fig. 2.2). This capacitance is connected in parallel to C e and is charged up to
U 0 aswell. Total charge of capacitors C e and CV can be expressed as
q0 = (Ce + CV ) U 0 .
(4)
Figure 2.2
Pressing switch K 2 connects the unknown capacitor C X in parallel to capacitor C e . Since
contacts of switch K1 are now open (forced by a spring), total charge of the system remains
constant. Due to redivision of charge between capacitors voltage falls from value U 0 to value
U 1 so that
q0 = (Ce + CV + C X )U 1 ,
or, taking in account formula (4):
(Ce + CV )U 0 = (Ce + CV + C X ) U 1 .
One can derive
U1 =
C e + CV
U0.
C e + CV + C X
(5)
Part of the systems charge remaining on plates of unknown capacitor is neutralised when
switch K 2 is depressed since it shorts the capacitor (see figure 2.2). Remaining charge on
capacitors Ce + CV can be expressed: q1 = (Ce + CV ) U 1 . Pressing switch K 2 again forces
part of charge q1 from capacitor C e to flow to capacitor C X again charge of capacitor
Ce + CV is reduced to value q 2 = (Ce + CV ) U 2 .
Here
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
U2 =
C e + CV
U1
C e + CV + C X
or, taking in account formula (5):
 C e + CV
C e + CV
U2 =
U 1 = 
C e + CV + C X
 C e + CV + C X
2

 U 0 .

(6)
In a similar way one can calculate voltage U n displayed by voltmeter V if capacitor C X is
pulsed by switch K 2 n times:
 C e + CV
C e + CV
Un =
U n −1 = 
C e + CV + C X
 C e + CV + C X
2

 C e + CV
 U n − 2 = ... = 

 C e + CV + C X
n

 U 0 .

One can derive capacitance of unknown capacitor C X as follows:

U
C X = (C e + CV )  0
 U n



 − 1 .



1
n
(7)
Formula (7) is correct only in case charge of C e decreases only due to charging and
discharging of capacitor C X . Condition is true when unwanted but always present resistance
Re (figure. 2.2) – which is total resistance of isolation of capacitor C e , switches K1 , K 2 and
wiring – is big enough. In that case (during measurements) charge coming through resistor
Re , is many times smaller than charge going to capacitor C X . In present experiment stand
value of Re is sufficient and causes and additional error no bigger than 1% if measurements
of C X are performed quicker than in 5 minutes.
Electrical diagram on experiment stand is shown on figure 2.3. C e is reference capacitor, C X
capacitor with unknown capacitance and CV – capacitance of voltmeter V.
Figure 2.3
4. Experimental procedure
1. Record the data of measuring devices and values of C e , CV and d.
2. Ask instructor to check connections and give an exact task.
3. Connect an electrostatic voltmeter (or standard multimeter – instructor tells which one) to the
stand. Switch the stand ON. When using a multimeter turn its mode switch to „ – V”.
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
4. Charge reference capacitor C e up to voltage U 0 (no more than 10,0 V) by pressing switch K 1
(figure 2.3). If voltage is higher than 10,0 V, ask instructor to adjust it. Write value U 0 to table
2.1.
5. Press and release switch K 2 . This connects parallel to C e the unknown capacitor C X , which has
air as dielectric and capacitance is C X 0 . Reading of voltmeter V falls somewhat, write it down.
Observing voltmeter readings and pressing and releasing switch K 2 reach the situation when
voltage reading is at least 2/3 of the original reading. Count pressing/depressing events n of switch
K 2 together with voltmeter readings U n and record them to corresponding columns of row
C X 0 of table 2.1.
6. Repeat the experiment at least 5 times. Values of U 0 , U n and n may be different in different
experiments.
7. Repeat the experiment with one plate inserted to unknown capacitor (its capacitance is then C X 1 ).
Then repeat experiment with 2 plates inserted. Value of unknown capacitor is C X 2 . Record all
measurement data to corresponding rows of table 2.1.
Table 2.1
Determining capacitance of unknown capacitor
Capacitance
of measured
(unknown)
capacitor
Expe
rim.
nr
U 0 ,V
n
Un ,
V
(C
C X i ,pF
Xi
− CX
)
2
1.
...
CX 0
5.
CX 0 =
...
±
...
1.
...
CX1
5.
CX1 =
...
±
...
CX 2 =
…
±
...
1.
...
CX 2
5.
C e = ...
C V = ...
d = ...
d’ = ..., ..., ..., ..., ...
d ′ =...
Voltmeter: .......
8. Use formula (7) to calculate for every measurement: C X i , mean C X for every capacitor and its
A-type modifier uncertainty on level 0,95.
9. Measure thickness of dielectric plates d’ in 5 different places. Calculate d ′ and it's uncertainty on
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TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS
level 0,95.
10. Using values C X 0 , C X 2 , d and d ′ and formula (3) calculate dielectric constant of plates ε . You
know that: C 0 = C X 0 and C = C X 2 .
11. Find the combined uncertainty U c (ε ) for ε knowing that d ≈ d ′ .
12. Knowing now value of ε , determine material of the dielectric plate.
13. Since you determined total capacitance C X 1 of 2 capacitors connected in parallel also in case there
was air between electrodes of one of them and dielectrical plate in another, one can write:
CX0
assuming that d ≈ d ′ . To check correctness of value ε calculated, make
2
certain that last formula is valid. Assume C X 0 = C X 0 and C X 1 = C X 1 .
C X 1 ≈ (ε + 1)
5. Questions and tasks
1.
2.
3.
4.
5.
Define the concept of capacitance. What parameters affect it?
Define the concept of dielectric constant of the insulating material using the Coulomb' law.
Substantiate the formula for calculating capacitance of parallel-plate capacitor.
How is charge distributed when two capacitors are connected in parallel?
What are the benefits of multiple charging method of unknown capacitor, used in this experiment
when compared to only one charge cycle (in this case C X is determined from formula (7) where
n=1)?
6. Why must the total resistance Re shown on figure 2.2. be as big as possible?
7. An empty (uncharged) capacitor is connected to a capacitor charged to some voltage. What
happens to systems total energy?
8. Number of times of pressing and depressing switch K 2 to discharge capacitor C e to given
voltage level is bigger in system with no dielectric plates between electrodes of measured
capacitor is bigger when compared to a setup with dielectric plates. Why?
9. Plates of a capacitor are shored with superconducting wire. What happens to energy stored in the
capacitor?
6. Literature
1. Halliday, D. Resnick, R., Walker, J. Fundamentals of Physics.–6th ed. New York, John Wiley &
Sons, Inc., 2001, §§ 26-1 – 26-4.
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