Lab Report №2: Capacitance
Objectives
● to study and observe properties of capacitors;
● to learn and observe the effect of the distance between the plates of a parallel plate
capacitor on its capacitance;
● to learn and observe the effect of a dielectric placed between the plates of a parallel plate
capacitor on its capacitance;
Experimental Data
D = 0.177 ± 0.0005 m (diameter of the plate)
ΔCe = not shown in the video (environmental error while no leads are connected)
ΔC1 = 2 * 10-11 F (systematic error when plates are far apart)
Δd = 0.15 mm = 0.00015 m (distance error set by manufacturer)
Table 1. Values of capacitance for separation between plates (Part 1).
Plate separation, d (m) Distance error, Δd (m) Capacitance, C (F)
Systematic error of
the capacitance, ΔC
(F)
0.001
0.00015
1.96 * 10-10
2 * 10-11
0.003
0.0005
0.86 * 10-10
2 * 10-11
0.005
0.0005
0.62 * 10-10
2 * 10-11
0.007
0.0005
0.51 * 10-10
2 * 10-11
0.009
0.0005
0.46 * 10-10
2 * 10-11
Table 2. Values of capacitance with paper used as a dielectric (Part 2).
Plate separation, Distance
d (m)
Δd (m)
error, Capacitance with Capacitance
dielectric, C (F) without
dielectric, C (F)
Systematic error
of
the
capacitance, ΔC
(F)
0.058
0.0005
6.1 * 10-11
4.3 * 10-11
2 * 10-11
0.06
0.0005
6.2 * 10-11
4.3 * 10-11
2 * 10-11
0.058
0.0005
6.2 * 10-11
4.3 * 10-11
2 * 10-11
Table 3. Values of capacitance with plastic used as a dielectric (Part 2).
Plate separation, Distance
d (m)
Δd (m)
error, Capacitance with Capacitance
dielectric, C (F) without
dielectric, C (F)
Systematic error
of
the
capacitance, ΔC
(F)
0.001
0.00015
2.28 * 10-10
2.09 * 10-10
2 * 10-11
0.001
0.00015
2.24 * 10-10
2.15 * 10-10
2 * 10-11
0.001
0.00015
2.26 * 10-10
1.94 * 10-10
2 * 10-11
Data Analysis
Formula derivation
Part 1.
The parallel-plate capacitor’s capacitance can be determined with formula below:
𝐴
𝐶 = 𝜀0 ∗ 𝑑
(1)
Where
𝜀0is a constant physical value known as the permittivity of free space. The theoretical value
of the permittivity is equal to the 8.85E-12 N/(C*m^2).
A defines the area of the one plate of the capacitor, the standard unit of the A is square
meter (m^2).
d is a distance between the plates of the capacitor, the standard unit is meter (m).
From the equation (1), we can see the relation of the area of the plate to the capacitance of the
parallel plate capacitor. The next step is to calculate the area of the one plate of capacitor. The
plate is a shape of the disk. Therefore, the area of the plate of the capacitor could be derived and
represented as:
𝐶=𝐶∗
𝐶2
4
(2)
Where
D is a diameter of the one plate of the capacitor.
𝐶 is a mathematical constant and has a value of 3.14159265359.
As a last step, the permittivity of free space (𝜀0) should be identified from the graph of capacitance
and area/distance. However, we can derive the formula from the equations (1) and (2), 𝜀0 can be
represented as :
𝐶
𝜀0 = 4𝑑 ∗ 𝐶∗𝐶2
(3)
Part 2
The relationship of the capacitances of the conductor with and without dielectric can be
represented as:
𝐶 = 𝑘𝐶0
(4)
Where
C is the capacitance of the conductor with dielectric.
𝐶0 is the capacitance of the conductor without dielectric.
k is the relative permittivity of dielectric material between plates.
The formula of the relative permittivity of dielectric material (k) derived from equation (4) is
𝐶
𝑘=𝐶
(5)
0
Error propagation
Part 1
The error of the area derived as:
𝜟𝐴 = 𝐶𝐷 ∗
𝜟𝐷
𝟐
(6)
The error of the permittivity of free space is
𝛿𝜀0
𝛿𝜀0
𝛿𝜀0
𝜟𝜀0 = √𝛥𝐶 2 ( 𝛿𝐶 )2 + 𝛥𝑑 2 ( 𝛿𝑑 )2 + 𝛥𝐴2 ( 𝛿𝐴 )2
Or
1
𝜟𝜀0 = 𝐴 √𝛥𝐶 2 𝑑 2 + 𝐶 2 𝛥𝑑 2 + 𝛥𝐴2 𝐶 2 𝑑 2
(7)
Part 2
The error of the relative permittivity of dielectric material (k) is
𝛥𝐶
𝛥𝐶
𝜟𝑘 = √( 𝐶 )2 + ( 𝐶 0 )2
0
Calculations and Visual representation
Part 1
The value of the area of the one plate:
The error of the area:
𝐶 = 0.0246 𝐶2
−4
𝛥𝐶 = 1.3902 ∗ 10 𝐶2
Figure №1. Relationship between Capacitance and the ratio of the area to the distance.
(8)
The slope here is 0.0685*10^(-10) C/(N*m^2) or 6.85*10^(-12) C/(N*m^2), which compared to
the theoretical of the permittivity of free space will be:
6.85*10^(-12) C/(N*m^2) < 8.85*10^(-12) C/(N*m^2)
Part 2
The relative permittivity of the dielectric material (k) for the paper stack:
𝐶 = 1.426
The error of the relative permittivity of the dielectric material (k) for the paper stack:
𝛥𝐶 = 0.567
The relative permittivity of the dielectric material (k) for the plastic sheet:
𝐶 = 1.097
The error of the relative permittivity of the dielectric material (k) for the plastic sheet:
𝛥𝐶 = 0.1314
Discussion
In the 1st part of this experiment, we tried to demonstrate the relationship between the
capacitance and the distance between the plates. In order to do that, we first measured the diameter
of one plate, and thereby derived its area. As you can see from Figure №1 we plotted the
capacitance in relation to the ratio of area to distance, the equation of which, as a result, gives us
the slope that we define as an experimental value of the permittivity of free space. Of course, the
difference between the theoretical and experimental values is considerable:
6.85*10^(-12) C/(N*m^2) < 8.85*10^(-12) C/(N*m^2)
However, it was expected in a sense, because there could have been many errors like systematic
errors and random errors that affected this: errors of the tools, multimeter’s errors and the error of
the distance between plates of the capacitor.
In the 2nd part of the experiment, we found the dielectric constants of the paper and the
rubber that we have worked with, as you can see from above in the Calculations part. Of course,
errors discussed in the previous paragraph also affected these measurements and calculations too.
Conclusion
To sum up, this experiment was carried out successfully, and thereby all objectives were achieved.
We understood how to work with multimeters and capacitors and observed it ourselves how the
distance or dielectrics affect the capacitance. As expected, no experiment can be conducted ideally,
and this particular experiment is no exception, since there were many errors that affected the final
results. However, it still did not prevent us from getting the best from this laboratory work.
Reference list
[1]
S. J. Ling, J. Sanny, W. Moebs. (2016). University Physics Volume 2 [Online]. Available:
https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2