LD
Physics
Leaflets
Electricity
Electrostatics
Plate Capacitor
P3.1.7.3
Determining the capacitance of
a plate capacitor
Measuring the charge
with the I-Measuring Amplifier D
Objects of the experiment
g Measuring the charge as function of the applied voltage
g Measuring the charge as function of the plate spacing
g Determination of the capacitance
Principles
If an electric conductor carries the charge +Q and on another
electric conductor is charged by the charge –Q, then a potential difference U exists between these conductors:
Q = C ⋅U
(I)
The proportionality factor C is called capacitance. The capacitance C depends on the geometrical arrangement of the
conductors and the − non conducting − material between
these conductors. The arrangement of the conductors is
called capacitor. The simplest design of a capacitor is that of
two parallel plates. The capacitance of a plate capacitor is
given by
C = εr ⋅ ε 0
A
d
εr = 8.85 ⋅ 10 −12
(II)
As
permittivity of free space,
Vm
εr: permittivity due to the material
+
+
+
+
+Q +
+
d: distance between the plates
εr
-Q
d
Bi 0106
A: area of the plates
A
A
U
Equation (II) holds only as long as the distance between the
plates is much smaller than the dimensions of the plates and
the electric field E between the plates can be considered to
be homogeneous. The permittivity εr describes the change of
the capacitance relatively to the vacuum value caused by the
introduction of the material.
In this experiment relation (I) is studied. The charge is measured by the I-Measuring Amplifier D as function of the applied
voltage U for various plate distances d. The capacitance C is
then determined as the slope of the straight line through the
origin and through the data points. Additionally, the variation
of the distance d between the plates allows to confirm the
proportionality
C∝
1
d
LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: info@ld-didactic.de
©by LD Didactic GmbH
Printed in the Federal Republic of Germany
Technical alterations reserved
P3.1.7.3
LD Physics leaflets
-2-
Carrying out the experiment
Apparatus
a) Measuring the charge as function of voltage
1 Parallel plate capacitor ...................................... 544 22
1 Tube power supply 0 to 500 V ........................... 521 65
1 Two-way switch ................................................. 504 48
1 I Measuring amplifier D...................................... 532 00
1 Multimeter LDanalog 20..................................... 531 120
1 Multimeter LDanalog 30..................................... 531 130
1 Measuring resistor 100 MOhm........................... 536 221
3 Pair cables 50 cm, red/blue ............................... 501 45
1 Pair cables 100 cm, red/blue ............................. 501 46
1 Connecting Lead 50 cm Red ............................. 500 421
-
Set the plates of the capacitor d = 2 mm apart.
To charge the plates apply a voltage of 50 V to the setup
as depicted in Fig. 1.
- Ground the I-Measuring amplifier D with the zero button 5
(see Fig. 1 and also instruction sheet 532 00).
- Disconnect the red cable from the fixed plate and connect
the coaxial cable BNC/4mm to the plate (Fig. 2).
Note: Instead of disconnecting the red cable from the fixed
plate and connecting the coaxial cable BNC/4mm in its place
an alternative setup may be used under certain circumstances. This setup is described in chapter supplementary
information.
- Measure the output voltage of the I-Measuring amplifier D
and determine the charge. Repeat this step several times
and determine the average values.
- Repeat the measurement for other voltages, e.g. 100 V,
150 V, 200 V, 250 V and 300 V.
- Repeat the measurement series for different plate distances d.
b) Measuring the charge as function of plate spacing
-
Setup
Set the plates of the capacitor d = 2 mm apart.
To charge the plates apply a voltage, e.g. 100 V, to the
setup depicted in Fig. 1.
Ground the I-Measuring amplifier D with the zero button 5
(see Fig. 1 and also instruction sheet 532 00).
Disconnect the red cable from the fixed plate and connect
the coaxial cable BNC/4mm to the plate (Fig. 2).
Measure the output voltage of the I-Measuring amplifier D
and determine the charge. Repeat this step several times
and determine the average values.
Repeat the measurement for the plate distances 4 mm,
6 mm, 8 mm, 10 mm and 12 mm.
-
Set up the experiment as shown in Fig. 1.
Note: The movable plate is connected to the earth socket of
the power supply and the I-Measuring amplifier D. The isolated plate is connected via the resistor 100 MΩ to the positive socket of the power supply.
For measuring the charge on the capacitor the I-Measuring
Amplifier D is switched to the range 10-8 As. The 3 V or 10 V
DC range may be selected at multimeter. Then, for example,
an output voltage of 3 V corresponds to 3⋅10-8 As.
A
V
-6
-7
10
1
10
10
0
10
10
-10
10
-1
-11
10
-2
10
10
A
-6
2
10
-9
10
-7
-8
-9
5.5
6
20
6.5
30
200
300
7
10
40
100
400
U
I,Q
1V(max.10Vs)
7.5
0
50
0
-7
10
2
10
10
-8
10
1
10
10
-9
10
0
10
4.5...7.5V
0...50V
0...500V
1A
5A
10mA
50mA
~
-
+
-
+
0
-10
10
-1
-11
10
-2
-6
10
-7
-8
-9
5.5
-
6
7
4.5
U
I,Q
+
1V(max.10Vs)
100MΩ
521 65
20
6.5
5
30
10
300
200
40
100
400
0
500
6.3V
532 00
As
-6
10
10
0
V
10
10
5
4.5
5
-
10
10
-8
10
-
As
0
10
-
7.5
0
50
0
500
6.3V
4.5...7.5V
0...50V
0...500V
1A
5A
10mA
50mA
~
-
+
-
+
-
+
100MΩ
521 65
532 00
OFF
O FF
531 120
OFF
531 120
V
3
100µ
1m
10m
100m
10
30
100
300
V
100µ
1m
10m
100m
300
1
300
3
531 120
3
1300m
100m
1m
100µ
100m
10m
DC
A
3
1
1m
300m
100µ
100m
100m
10m
3
100µ
1m
10m
100m
10
30
100
Fig. 1: Experimental setup (wiring diagram schematically) for charging the capacitor.
300
1
300
3
3
30
10
1
3
1300m
V
AC
3
100
AC
100m
1m
100µ
100m
10m
DC
A
1
3
1
V
A
3
30
10
DC
A
100µ
1m
10m
100m
A
1
300
3
10
30
100
100
1
30
10
V
300
V
1
30
10
V
531 120
OFF
AC
3
3
100
AC
3
100
V
A
A
1
300
3
10
30
100
1m
300m
100µ
100m
100m
10m
DC
A
Fig. 2: Experimental setup (wiring diagram schematically) for measuring the charge on the capacitor.
LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: info@ld-didactic.de
©by LD Didactic GmbH
Printed in the Federal Republic of Germany
Technical alterations reserved
LD Physics leaflets
P3.1.7.3
-3-
90
Measuring example
d = 2 mm
80
70
charge Q / nAs
a) Measuring the charge as function of voltage
Table 1: Charge Q (average over 3 measurements) as function of the applied voltage U for fixed plate distances d.
U
V
Q
nAs
Q
nAs
Q
nAs
(d = 2 mm)
(d = 4 mm)
(d = 6 mm)
60
50
4 mm
40
30
20
6 mm
10
50
12.0
6.8
4.5
100
24.0
13.0
9.0
150
35.5
19.7
13.9
200
48.0
26.5
18.3
250
61,5
33.3
23.0
300
74.0
40.0
27.9
0
0
50
100
150
200
250
300
voltage U / V
Fig. 3: Charge Q as function of the applied voltage U for various
plate distances d The straight line corresponds to a fit
through the origin according equation (I).
b) Measuring the charge as function of plate spacing
b) Measuring the charge as function of plate spacing
Table 2: Charge Q (average over 3 measurements) as function of the plate distances d for the applied voltage U = 100 V.
The capacitance is determined by equation (I).
C
pF
2
24.0
240
4
12.8
128
6
9.0
90
8
7.1
71
10
6.0
60
12
5.2
52
250
200
capacitance C / pF
Q
nAs
d
mm
150
100
50
0
0
2
4
6
8
10
12
plate distance d / mm
Evaluation and results
Fig. 4: Capacitance C as function of the plate distances d for
U = 100 V.
a) Measuring the charge as function of voltage
Fig. 3 summarizes the result of table 1. The straight lines
correspond to fits according equation (I). From the slopes the
capacitance is determined to:
C = 244 pF
(d =2 mm)
C = 133 pF
(d =4 mm)
Fig. 4 summarizes the result of table 2. In accordance with
equation (II) the capacitance decreases with increasing plate
distance d.
In Fig. 5 the capacitance is plotted versus the inverse values
of the plate distance d to confirm the proportionality:
C∝
C = 92 pF
1
d
(d =6 mm)
The deviations are due additional capacities in measuring
setup and an increasing region of the inhomogeneous electric
field on the bothers of the plates.
LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: info@ld-didactic.de
©by LD Didactic GmbH
Printed in the Federal Republic of Germany
Technical alterations reserved
P3.1.7.3
-4-
LD Physics leaflets
250
capacitance C / nF
200
150
100
50
0
0,0
0,1
0,2
0,3
d
-1
/ mm
0,4
0,5
-1
Fig. 5: Capacitance C as function of plate distances d The straight
line corresponds to a fit through the origin according equation
(I).
Performing a fit according equation (II) the permittivity ε0 (εr ≈
1) can be determined form the slope (Fig. 5). Taking into
account the inevitable additional capacities (order of magnitude 15 pF) the slope has been determined to:
Slope: s = 451⋅pF⋅mm
-2
2
Area of the plates: A = 5.1⋅10 m
s
-12 As
= 8.83⋅10
A
Vm
ε0 =
Supplementary information
Using the two-way switch (504 48) the two steps charging the
capacitor (Fig. 1) and measuring the charge on the capacitor
(Fig. 2) can done in one step conveniently. However, this
setup depicted in Fig 6 may not work properly at ambient
conditions with high humidity due to leakage currents.
A
10
V
-6
10-7
As
0
102
10
-6
10-7
10
-8
10
1
10
-8
10
-9
10
0
10
-9
10-10
10 -1
-11
-2
10
10
100MΩ
5.5
6
20
6.5
5
7
30
10
200
40
300
100
400
0
4.5
U
I,Q
1V(max.10Vs)
7.5
0
50
0
500
6.3V
4.5...7.5V
0...50V
0...500V
1A
5A
10mA
50mA
~
-
+
-
+
-
+
521 65
532 00
OFF
531 120
OFF
531 120
V
3
100µ
1m
10m
100m
10
30
100
300
V
100µ
1m
10m
100m
300
AC
1300m
V
AC
1
30
10
3
3
3
100
3
3
100
1
300
100m
1m
100µ
100m
10m
DC
A
1
30
10
3
1300m
V
A
A
1
300
3
10
30
100
100m
1m
100µ
100m
10m
DC
A
Fig. 6: Alternative experimental setup (wiring diagram schematically), compare Fig. 1 and Fig. 2, respectively.
LD Didactic GmbH . Leyboldstrasse 1 . D-50354 Huerth / Germany . Phone: (02233) 604-0 . Fax: (02233) 604-222 . e-mail: info@ld-didactic.de
©by LD Didactic GmbH
Printed in the Federal Republic of Germany
Technical alterations reserved