Lab II: Electric Field Mapping
George Wong
Jenna Snyder
Instructor: Geoffrey Ryan
Experiment Date: 14 February 2012
Due Date: 28 February 2012
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Objective
The primary objective of this laboratory examination was to work with and experimentally
observe the relationship(s) between potential voltage differences and electric fields produced
by charged conductors etc..
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Theory
The forces that manifest between charged particles is connected to the electric field construct
~ Some charge q experiences a force of q E
~ in such an electric field.
E.
Voltage
potential between two points is given as Va − Vb = Vab . This can also be written
Rb
~
as a E · d~l. Further, a volt is defined to be “the difference in electric potential across a
distance”.
A voltmeter measures the voltage difference between two different points. These points are
identified by making electrical connections. A line connecting points of equal voltage is called
an equipotential.
Electric field lines and equipotentials are always completely orthogonal to each other. Also,
~ density is directly proportional to the density of the equipotential lines.
the E
It might be noted that while voltage differences do exist in free space, voltmeters cannot
measure these differences and thus in experiments, we must use physical analogues. Considering this, we may note that working in two dimensions is essentially equivalent to working
in three dimensions, if we imagine the two dimensional diagrams extruded out along some
vertical z dimension.
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Set Up
The main device that was used for this lab was a mapping board setup and accompanying
resistive boards. The mapping board allowed for various voltages to be ‘generated’ and ran
through the various resistive boards, connected at low-resistance electrodes. Included with
the board was a section of resistors in series (so as to allow for discrete-drops in voltage;
a ‘u’-probe to measure voltage at any certain point; a voltmeter (to read the voltage); and
specific-design resistive boards.
The resistive boards were essentially insulated plates coated with a highly resistive material having discrete sections coated with low-resistance materials (such as metals) acting as
electrodes. The electrodes on these plates were of various configurations, shown below.
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Figure 1: Mapping-device Setup
Figure 2: Resistive Plates
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Procedure
Starting with board (3)
1. Resistive board was attached to the mapping apparatus.
2. Voltage supply was attached to binding posts and ‘u’-probe, set to low level by selecting
a specific resistancetotal peg.
3. The probe was dragged across the resistive board until the voltmeter read 0, at some
point.
4. At this point, the a mark was made, signifying a specific-voltage point.
5. The previous two steps were repeated many times until a good collection of equalpotential points was had.
6. These points were all connected to create a equipotential line.
7. The previous five steps were repeated for various resistance levels (as in by changing
the resistancetotal peg).
8. Board was swapped out for: (3) → (4) or (4) → (5).
9. Materials Replaced; Station Tidied
10. Data Analyzed
2
Figure 3: How to Measure Voltage Difference
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Data
See diagrams attached at the end of this report.
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Calculations
No direct numerical calculations were done over the course of this lab. The closest process
to calculations might be considered to be guessing where the equipotential lines should be
drawn to properly connect the measured points. Similarly, the decisions regarding drawing
~ field lines might also be considered to be a sort of calculation.
of E
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Error Analysis
Due to the peculiar nature of the “data” taken for this laboratory examination, the error
analysis will not so much consist of numerical studies etc., but rather will deal with the way
that error manifested itself and presented itself qualitatively.
Obviously, in the drawing process, error resulted from imprecise marking of the points where
voltage was supposed to be equal. This undoubtedly came about as a result of parallax error
(in looking at the needle on the voltmeter), pencil thickness, improper calibration of the
probe and/or voltmeter, and any random error in voltage reporting.
Further, when the equipotential lines were being drawn, error was undoubtedly had as the
points were connected. There was a good amount of guesswork involved in choosing where to
draw the lines/how to connect the points. While the equipotentials were drawn to relatively
reflect what was assumed to be proper locations, such could not necessarily be done with
perfect precision.
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Questions
Section 2
Can you guess in advance what happens? Will the field lines pass through the bodies, or go
around them, or what?
My guess would be that the field lines will appear as they normally do (seemingly ‘generated’
about the primary electrodes); however, that the insulator and conductor will appear to warp
them slightly, such that the field lines somewhat bend towards or away from the ‘foreign’
objects. I do not believe that any voltage potentials will exist within either the insulator or
the conductor.
Section 4
1. Where are the equipotential lines closest together and where are they farthest apart?
Generally, the equipotential lines were closest together near the densely focused electrodes
(as in the circle), whereas they were farthest apart when farthest from the electrodes (as in in
the intermediate space between various conductors/insulators)—this being especially shown
on the plot of Pattern 3 in the middle space between the two ‘electrodes’.
2. Do any of your equipotential lines cross? Do you think it is possible for equipotential lines
or surfaces to touch or cross?
No equipotential lines cross, neither is it possible for lines (or surfaces) to touch or cross.
Were it the case that this were possible, that would mean that there would be two different
voltage potentials at the same point in space. As the potential ‘field’ is a scalar field, this
makes no sense. There is no way that a location could have two distinct potentials at the
same time.
3. What is the direction of the equipotential lines near the edges of the electrodes?
Near the electrodes, the equipotential lines were essentially completely parallel to the edges
of said electrodes. This parallelism increased as the distance to the edge decreased.
Section 5
1. If the polarities of the electrodes are reversed, how do the patterns of electric potential and
field change?
Were the polarities reversed, the patterns would (presumably) stay the same, but the direction
of the fields would be opposite (anti-parallel at all points).
2. Is it possible for electric field lines to cross? Explain.
~ field lines to cross. Were this possible, it would indicate that the
It is not possible for E
vector at that point would have two (or more) different directions, which is nonsensical. As
~ field is a vector field, each point in space has some vector attribute, but it does
the E
not—cannot—have multiple vector values. Were lines to cross, there would need to be two+
different vectors at one point (one for each of the lines participating in the crossing-action.
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3. What feature of the equipotential map indicates where the electrical field is strongest?
Why? Is the electric field strongest where you expect it to be?
The higher-density parts of the equipotential maps indicate where the electrical field is
~ field lines drawn at each point is proportionally related to
strongest. The number of E
the the density of the equipotential field lines. Assuming a normal distribution (as should be
the case), there is a direct relationship between the number of lines in a certain area and the
way that lines perpendicular to the aforementioned lines are angled. The greater the angles,
the greater the spreading. Greater angles also suggest a greater spread of the perpendicular
~ field is indeed strongest where it would be expected to be strongest: near the
lines. The E
electrodes (also, the ‘denser’ the electrode, the stronger the field.
~ perpendicular to it is an
4. Explain in your own words why a surface that always has E
equipotential.
We can think of the electric field as being an infinitely dense vector field; we can say that
every point in space has a direction associated with it. If we take each point in space and
find the directions orthogonal to it and connect each of these infinitely small perpendicular
planes, we would end up with surfaces that are constantly perpendicular to the electric field.
As these two are being defined (mostly) as being perfectly orthogonal, we see that such a
relationship would necessitate such a relationship between the two. The surfaces formed
~ and thereby be equipotentials.
would be perfectly perpendicular to E
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Conclusion
Overall, the investigation did quite facilitate the purpose of having the student examiner
better acquaint him/herself with the relationships between electric fields and equipotentials
in terms of voltage difference potentials. Of course, the nature of this laboratory examination
necessarily meant that the results we—the experimenters—got were mainly qualitative (as
opposed to quantitative).
~ and voltage equipotentials were perpendicular to
Generally, we were able to observe how E
~ density
each other. Further, we were able to determine the graphical relationships between E
and equipotential density.
Of course, various previously-assumed ‘assumptions’ were experimentally/observationally
confirmed: equipotentials run parallel to the conducting field-inducers (electrodes). Such
can be seen on the attached diagrams (with field lines drawn in).
In future experiments, it might be an interesting pursuit to increase the resolution of the
equipotentials (with the use of more resistors, or a differently-stated methodology for identifying points of equal voltage). While the number of lines used in the current rendition of
the lab allows for basic concepts to be understood, a more ‘complete’, fuller picture might be
had were more lines to be drawn. Along the same lines (no pun intended), a more accurate
means of drawing the exact curvature of the lines might allow for more interesting pictures
to be made.
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Of course, more ‘interesting’ resistive plates would also, theoretically, provide the examiner
with a more interesting experiment. Besides complicating the patterns made by the electrodes, it might be an interesting pursuit to develop plates with varying-property insulators
and conductors irregularly spaced and of varied patterns in design.
The more complex plates in combination with higher-resolution equipotential lines would
theoretically allow for the student to see better how complicated relationships can get and
also to see the effect that insulators of varying properties have on the field lines.
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