Jason Fong
702847140
Lab Section 3
Partner: Kiyomi Tsuruta
Exp Date: 1-23-00
Physics 4BL: Experiment 2
This experiment explored the configurations of electric fields. In particular, the
electric fields for two parallel plates and two line charges were investigated. Both of the
configurations were represented two-demonsionally on a piece of conductive paper.
Equipotential lines for each of the configurations were plotted, and then the field lines
were drawn by following a path othogonal to each of the equipotential lines.
The parallel plates were represented by two metal bars placed on a sheet of
conductive paper. A voltage of 10 volts was applied to one bar, and the other was left at
0 volts. A Fluke multimeter was used to measure the voltage at points on the paper.
Points were plotted for voltages at one volt intervals from zero to ten volts. Lines
connecting points of the same voltages gave the equipotential lines of the parallel plates.
These lines ran roughly parallel to the plate in the region between the middles of the
plates, and the curved outward as they approached the ends of the plates. Another set of
lines were then drawn to represent the electric field between the plates. These lines
followed a path othogonal to each of the equipotential lines since electric field lines are
always orthogonal to equipontential lines. The electric field lines were fairly straight in
the region between the middles of the plates, and curved out at the regions closer to the
edge of the plates.
The magnitude of the electric field was then calculated for four different points.
Two points were selected from the region between the plates, and two were selected from
the area not between the plates. The points between the plates are marked as points I and
II on the electric field lines diagram. The points outside of the plates are marked as
points III and IV. The magnitude of the electric field can be calculated by using the
following formula:
E
V V2  V1

S S2  S1
where ΔV is the difference in voltage and ΔS is the distance between two succesive
equipotential lines. The error in this calculation is given by this formula:
 E
  E 
1
  V 
E  
V    S    V    2 S 
 V
  S 
S
  S

2
2
2
2
where V and S correspond to ΔV and ΔS. Taking the error in the voltage measurements
to be δV = ±0.05V and the error in the distance measurements to be δS = ±2mm, the
following table was compiled for the magnitude of the electric field calculated at the four
points.
Point Magnitude of Electric Field E
I
120 ± 30 V/m
II
150 ± 50 V/m
III
48 ± 5 V/m
IV
42 ± 4 V/m
At the two points between the plates, points I and II, the magnitude of the electric
field should be the same if the configuration were a true infinite parallel plate. The
electric field is uniform between an infinite parallel plate configuration because only the
component of the electric field vectors directed straight at the other plate is significant.
The other components of the vectors are canceled by neighboring components. However,
since the plates used the experiment are finite and have ends, the electric field will not be
uniform. Near the ends of the plates, the components of the electric field which are not
directed straight at the other plate are not completely canceled since there is no opposing
field components on the open side of the parallel plates. Thus, the field lines will curve
outwards and cause the magnitude of the electric field to change. However, the two
points were selected in the inner regions of the plates in order to minimize the effects of
the curving field at the ends of the plates. The calculated values are not exactly equal
because the points are at different locations were the field curves in different ways due to
the plates not being infinite. There is also a systematic error in the measurements due to
the finite size of the conductive paper. The measurements were taken assuming that the
surface of the paper is uniform throughout. However, toward the edges of the paper, the
electric potential will behave differently so that the measurements will be less accurate
toward the edges. If we take into account the uncertainty due to the possible errors in the
measurements, we see that the two values between the plates are in agreement within the
calculated uncertainties. This suggests that the electric field would indeed be uniform
between the plates if the plates were very large.
The line charges were represented by two metal cylinders. One cylinder was set
at 10 volts, and the other at 0 volts. As with the parallel plates, points around the
cylinders were plotted for voltages at one volt intervals between zero to ten volts.
Equipointenial lines were drawn be connecting points of the same voltage, and electric
field lines were drawn by following paths orthogonal to the equipotential lines.
Taking R0 as the radius of the cylinders, and x0 as the distance from the center of
a cylinder to the midpoint between the cylinders, we can find the distance d from the
image line charge to the midpoint. The formula relating those values is:
x02  R02  d 2
The radius of the cylinders were measured to be R0 = 9mm and the distance from the
center of a cylinder to the midpoint between the cylinders was found to be x0 = 34mm.
Using those values, d can be found:
d  x02  R02  342  92  32.8mm
For each equipotential line, we can define a value γ such that for the conductors we have
2

x0  R0  d 
0 
x0  R0  d 2
and for the ith equipotential line, we have
i 
Vi
V0
0
with V0 = -5V.
Using these values, the theoretical values for the radii and centers of the equipotential
circles can be calculated. The radii are given by:
R
2d 
 1
The centers are given by:
xc 
d   1
 1
One the predicted values are calculated, they can be compared to the actual values
measured on the paper. The values measured on the paper were based on potentials of 0
and 10 volts on the cylinders. However, the predicted values are based on potentials of -5
and 5 volts. This difference can be solved by subtracting 5 from the measured values for
the voltage because the potential difference between the two cylinders will remain the
same. The following table summarizes the results of the predicted and actual values and
the differences between them.
0
1
2
3
4
5
6
7
8
9
10
-5
-4
-3
-2
-1
0
1
2
3
4
5
55.23
24.76
11.10
4.98
2.23
1.00
0.45
0.20
0.09
0.04
0.02
34.01
35.56
39.30
49.30
86.11
8.99
13.74
21.64
36.81
79.61
86.11
49.30
39.30
35.56
34.01
79.61
36.81
21.64
13.74
8.99
34
35
38.5
51
46
37
35.5
34.5
5
0.01
3.99
7.5
0.56
6.24
14
0.80
7.64
33
1.70
3.81
circle extends off page
no circle
circle extends off page
26
3.30
10.81
13
2.30
8.64
7
0.06
6.74
5.5
0.49
3.49
0.03
1.58
2.02
3.45
44.38
45.41
35.31
10.34
6.69
5.84
0.17
1.44
29.36
39.93
49.05
38.82
The centers of the equipotential circles agree fairly well with the predicted values.
The radii of the circles, however, are very far off from the predicted values. One possible
source of that error could be in the difficulty in drawing the circles accurately. The
circles were drawn based on a limited set of reference points, so the circles could very
easily be mishapened and have a different radius. Another source of error could be the
finite size of the paper. The predicted value assumes that the electric field is free to
spread out as far as it needs to go. But the conducting paper limits the area where the
equipotential lines can be observed. For instance, the -1 and 1 volt equipotential lines are
incomplete circles because they run off the edge of the paper. The different behavior at
the edges of the sheet could cause the equipotential lines to be more bunched together
toward the center of the paper, which would cause the radii to be smaller. As seen in the
chart, the radii of all of the circles are indeed smaller. Also supporting this line of
reasoning is that the centers of the circles are in roughly the correct position. This
suggests that near the middle of the paper, the equipotential lines behave more closely to
the predicted values, but as they spread out they become more restricted. This leads to
the smaller radii observed in the experiment.
This experiment explored the behavior of electric fields in a parallel plate
configuration and in a two line charge configuration. The plotted equipotential and
electric field lines did not behave exactly as predicted, but the differences can be
accounted for in the various sources of measurement uncertainties, the imperfect
configurations of the parallel plates and line charges, and the finite size of the conductive
paper. However, the experiment still gave valuable insight into the behavior of electric
fields.