# Objective constant can also be found with the combination

Objective

The
purpose of the experiment was to test the effect of dielectric material on a
calibrated parallel plate capacitor.

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Theory

Capacitors are devices made of two parallel plates separated by a
distance with equal area but opposite charges that can store potential energy
in an electric field. The capacitance, measured in farads F,  can be calculated using the following
relationship (Department of Physics):

(1)

where A
is the area of each plate, d is the
distance between the plates and  is the permittivity of the material between
the plates. This relationship illustrates that the capacitance is determined by
the size and shape of the conductors and the separation between them.
Furthermore,  as the distance between the
plates increase, the capacitance would expectedly decrease (Department of Physics). Capacitors are charged by having electrons removed from the positive
plate and carried to the negative plate which is directly against the electric
field to which in nature is pulling the electrons back to the positive plate
and pushing them away from the negative plate. Therefore, the capacitance can
be measured by charging the capacitor then discharging it through the
galvanometer in the circuit (What is dielectric
constant?).

The deflection of the galvanometer if proportional to the current
going through it. Therefore, when charging the capacitor to a potential
difference, V, the charge placed can
be charged as q, the capacitance can
be found using the following relationship (Department of
Physics):

(2)

It
is assumed that the permittivity of air is similar to vacuum, therefore the
permittivity of vacuum, is used. The ratio of relative to is called the dielectric constant, k,
which  interprets the ability of
the material to concentrate electric flux, therefore expressed in the following
equation:

(3)

The dielectric constant can also be found with the combination of
equation (1) and (3) resulting in the following equation:

(4)

where   is measured capacitance with vacuum between
the plates and C is the measured
capacitance with the dielectric material(Department of Physics).

Procedure

Part
A

1.
Apparatus was set up as shown in the
following diagram using the standard (known) capacitor:

2.
Figure (1): Illustration of the electric circuit
used to set up the apparatus to charge and discharge the capacitor (CITE)

3.
The sensitivity of the galvanometer was
calculated by setting the galvanometer range to “Direct” and charging the
standard capacitor to 66.7 V. The capacitor was allowed to charge for a few
seconds as it was charging through a large resistance.

4.
The maximum value of deflection was recorded by
the galvanometer on the first swing when switched to discharge and an average
reading was taken after a few trials.

5.
The standard capacitor was replaced with the
parallel plate capacitor. This was done carefully to ensure that the capacitor
plates just touch with no significant pressure exerted on each other.

6.
Ensuring that no calibration on the Vernier scale
was needed, the plate separation was set to 0.3 mm. The parallel plate
capacitor was then charged to  a starting
voltage of 66.7 V and the deflection was recorded. The charge was increased
when needed to provide a reasonable deflection on the galvanometer.

7.
Step five was then repeated for several plate
separation between 0.3 mm and 10.0 mm with smaller intervals for shorter
distances to provide better results.

8.
The area of the plate was then recorded.

9.
Six sheets of paper was then obtained and the
thickness of a single paper was measured.

10.
One sheet of paper was placed between the plates
and the plates were then closed on it until it was slightly snug but not tight.
A potential difference of 141.4V was used to obtain a deflection on the
galvanometer.

11.
Step nine was repeated to obtain results by
adding a single paper each time to obtain consecutive results up to six sheets.

12.
Steps eight through ten were then repeated for
two other materials to obtain results, however only obtaining results for.

13.
The galvanometer was then turned to “SHORT”. The
high voltage source and the galvanometer were then unplugged. (Department of
Physics).

Observations
and Results

Area  of the Parallel Plates:  0.0556 m^2=556cm^2

Known Capacitance = 8.07nF= 8.07E-09F

Part
A:

Table
1: Calibration of the sensitivity for the Galvanometer.

Value of
deflection (centimeter)

Average
Deflection from 3 (centimeter)

Average
Deflection Total
(centimeter)

Voltage(Potential
Difference)

Sensitivity

2.9±

3.03

2.965

66.7

1.82×10^-7

3.1

66.7

3.1

66.7

2.9

2.9

66.6

2.9

66.8

2.9

66.6

3.1

2.97

66.6

2.9

66.7

2.9

66.7

3

2.96

66.7

2.9

66.7

3

66.7

Calculations:

Uncertainty=(0.25)(Capacitance)
+ 0.5pF

Uncertainty=(0.25)(8.07nF)
+ 0.0005nF

Uncertainty=2.018nF

Sensitivity=
(Voltage)(Known Capacitance)/Average deflection

Sensitivity=
(66.7)(8.07×10^-9)/2.965

Sensitivity=
1.82X10^-7

Part
B

Table
2: Deflection values and capacitance values obtained for corresponding
plate

separation and potential difference.

Plate
Separation (centimeters)

Value of
deflection (centimeters)

Voltage
(Potential
Difference)

Capacitance
(F)

1/Separation
(centimeters)

.03

2.6

143.6

3.295×10^-9

33.3

.05

1.5

151.9

1.797×10^-9

20.0

.08

1.5

151.9

1.797×10^-9

12.5

0.1

0.7

151.7

8.398×10^-10

10

0.12

0.5

151.7

5.999×10^-10

8.33

0.15

0.3

151.7

3.599×10^-10

6.67

0.2

0.2

151.7

2.399×10^-10

5

0.25

0.1

161.3

1.128×10^-10

4

0.3

0.2

166.9

1.090×10^-10

3.33

0.35

0.2

166.9

2.181×10^-10

2.86

0.4

0.2

191.7

1.899×10^-10

2.5

0.5

0.1

199.5

9.49×10^-11

2

0.6

0.1

199.5

9.123×10^-11

1.67

0.7

0.1

199.5

9.123×10^-11

1.43

0.8

0.1

199.5

9.123×10^-11

1.25

0.9

0.05

199.5

4.561×10^-11

1.11

1.0

0.05

199.5

4.561×10^-11

1.00

Example
Calculations:

1/Separation=1/0.03

1/Separation=33.3

Capacitance=(sensitivity)x(deflection)/voltage

Capacitance=(1.82X10^-7)x(2.6)/143.6

Figure 2: Graph illustrates the capacitance
obtained corresponding to 1/separation

for air.

Capacitance=3.295×10^-9

?=Slope/Area

?=1×10^-10/556

?=1.799E-13
F/cm

?=1.799E-11F/m

2.03

(Ashby)

Error
propagation:

(1-2.03/ 1)= 103% error

Part
C

Table
3: Measurements of material thickness

Thickness of Paper

0.75mm

Thickness of material 1 – plastic

0.45mm

Thickness of material 2

0.4mm

Table
4: Deflection values obtained for corresponding number of sheets places between
the

plates.

Sheets of Paper

Deflection

Sheets of Material 1

Deflection

Sheets of Material 2

Deflection

Voltage (Potential Difference)

1

2.2

1

1.1

1

2

141.4

2

n/a

2

n/a

2

n/a

141.4

3

1.6

3

0.9

3

1.7

141.4

4

1.4

4

n/a

4

n/a

141.4

5

1.3

5

n/a

5

n/a

141.4

6

1.2

6

0.3

6

0.6

141.4

attained due to time constraint

Table 5: Separation values and the calculated
values for 1/separation.

Separation
(centimeter)

1/separation
Paper (1/centimeter)

Separation
(centimeter)

1/separation
Material-1 (1/centimeter)

Separation
(centimeter)

1/separation
Material-2 (1/centimeter)

0.075

13.33

0.045

22.22

0.04

25

0.15

6.67

0.09

11.11

0.08

12.5

0.225

4.44

0.135

7.41

0.12

8.33

0.3

3.33

0.18

5.56

0.16

6.25

0.375

2.67

0.225

4.44

0.2

5

0.45

2.22

0.27

3.70

0.24

4.17

Table 6:
Calculated Capacitance for Paper, Material 1, and Material 2.

Capacitance
Paper F

Capacitance
Material 1 F

Capacitance
Material 2 F

2.832×10^-9

1.416×10^-9

2.574×10^-9

n/a

n/a

n/a

2.059×10^-9

1.158×10^-9

2.188×10^-9

1.801×10^-9

n/a

n/a

1.673×10^-9

n/a

n/a

1.54×10^-9

3.861×10^-10

7.723×10^-10

Example
Calculation:

Capacitance=(sensitivity)x(deflection)/voltage

Capacitance=(1.82X10^-7)x(2.2)/141.4

Capacitance=
2.832×10^-9 F

Figure
3:Graph illustrates the capacitance obtained corresponding to 1/separation

for sheets of
paper.

?=Slope/Area

?=1.1×10^-0/556cm^2

?=1.978E-13
F/cm

?=1.978E-11
F/m

2.23

(Ashby)

Error
propagation:

(3.85-2.23/ 3.85) x 100= 42% error

Figure
4: Graph illustrates the capacitance obtained corresponding to
1/separation

for first
material used.

?=Slope/Area

?=4E-11/556

?=7.19E-14F/cm

?=7.19E-12F/m

0.812

(Ashby)

Error
propagation:

(2.5-0.812/ 2.5) x 100= 67.5% error

(assuming
this is wax paper)

Figure
5: Graph illustrates the capacitance obtained corresponding to
1/separation

for second
material used.

?=Slope/Area

?=7E-11/556

?=1.258E-13F/cm

?=1.258E-11F/m

1.421

Error
propagation:

(2.25-1.421/ 2.25) x 100= 36.9% error

(assuming
this is polyethylene)

Discussion
and Conclusion

Part
A and Part B:

As was seen in the results, the value of
permittivity of air (k)that was calculated for was2.03.  This is not close to the known permittivity
of k(air)=1.00058986.
Where the results attained had an inaccuracy of nearly 103%. The potential sources of error could include the fact
that galvanometer did not indicate an exact number, but was almost like a
pendulum in how the results were portrayed. Therefor measuring the exact
deflection amount relied heavily on what the experimenter thought the line was
at. This could have lead to the slight difference in the permittivity of the
air that was attained. However the graph (Figure 2), shows that the results
were as expected. Where there is a linear correlation between the Capacitance
and the 1 over the separation when looking at the line of best fit.

Part C:

As seen in the results and
calculations, the permittivity of the materials is as follows:

Paper

k=2.23

Material 1

k=0.812

Material 2

k=1.421

When compared to the
online permittivity of paper, the error percentage is equal to 42% . This error percentage could be explained by
potential sources of error like the fact that the paper itself was dirty
leading to a potential foreign material being introduced, therefore leading to
two different permittivities being introduced. This error is carried on through
the experiment for Material 1 and Material 2. The k value for material 1 and
2  both yielded a error propagation of
67.5% and 36.9% respectively. For part C other than the fact that the materials
were all dirty, another source of inaccuracy could potentially have arisen from
the fact that galvanometer was in itself an inaccurate device. Similar to part
A and B, the results are difficult to read leading to some guess work as to
where the maximum line had reached.

Through the potential inaccuracies, the graphs (figure 2,3,4,5)
show a linear correlation as expected between capacitance, and
1/separation.  This correlation is what
has allowed for the calculation of the permittivity.