Download LAB 3 Equipotential Mapping

LAB 3
Equipotential Mapping
OBJECTIVES
1. Determine the location of equipotential surfaces in the region around various
electrode arrangements.
2. Construct electric field lines from the measured equipotential surfaces.
EQUIPMENT
Field mapping apparatus, electrode plates, templates, paper, DC Power Supply, digital
multimeter
THEORY
Consider two electrodes of arbitrary shape some distance apart carrying equal and
opposite charges. There will then exist a fixed potential difference between the
electrodes. Suppose that this potential difference is 15 V. If the electrode with the
negative charge is arbitrarily assumed to be at zero potential, then the electrode with
the positive charge is at a potential of +15 V Given these assumptions, in the space
surrounding these electrodes there will exist points that are at the same potential. For
example, for the case described above, there exist some points for which the potential
is +9 V other points for which the potential is +12V, and still other points for which the
potential is +6 V. In a three-dimensional space, all points at the same potential will
form a surface, and there will be a different surface for each value of the potential
between 0 V and 15 V. In fact, there exist an infinite number of such surfaces because
one could divide the 15-V total potential difference into an infinite number of steps.
Each of these surfaces with the same value of potential is called an equipotential
surface. In this laboratory, the equipotentials for a few simple, but often used,
electrode configurations will be determined.
In addition to the equipotential surfaces that exist in the region around charged
electrodes, an electric field is also present. By definition, the electric field is a vector
field, which can be represented by lines drawn from the positively charged electrode to
the negatively charged electrode. The electric field lines must always be perpendicular
to the equipotential surfaces. This can be derived from the fact that it takes no work to
move an electric charge on an equipotential surface because along the surface, ∆V =
0. If the work is zero, then the electric field along the equipotential surface must be
zero.
A direct-current power supply will provide the source of potential difference and will
serve to keep the voltage between the two electrodes fixed at whatever value is
chosen from the power supply. The electrode to which the negative terminal of the
power supply is attached will arbitrarily be chosen to be the zero of potential, and all
measurements will be made relative to that electrode. A voltmeter will then be used to
find the points on the paper in the region of the electrodes that are at some given value
of potential. Once enough points at that potential have been located to establish the
shape of the equipotential, the equipotential line can be constructed by joining the
points with a smooth curve.
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PROCEDURE
Field Mapping for different Electrode Arrangements
a. Mount a conducting plate on the bottom of the field mapping table and a piece of
graph paper on top. Use the template to draw your conductor configuration on the
graph paper.
b. Connect the power supply to the field-mapping table and adjust the output voltage of
the power supply to 15 V using your voltmeter. This will produce a small current that
flows through the slightly conducting coating on the board.
c. Connect the black lead of the voltmeter to the ground terminal of the power supply.
To explore the voltages on the conducting board, you will measure voltage
differences or potential differences between your voltmeter’s red and black leads.
The black lead is always kept at the ground terminal of the power supply and the red
lead will be moved from one location to another in order to measure different
potentials. When a series points have the same potential, it is called an equipotential
line.
d. In order to make a contour map of equipotential lines on the paper, connect the V-Ω
lead from the meter to the probe on the field-mapping board. Systematically search
for a number of points whose potential is about 9 V. Mark them, and draw a smooth
line connecting them (don't connect the dots with straight lines!). Then repeat this
process for increments of 1.5V for equipotentials ranging between 3V through 12V.
Each smooth line is the “equipotential” for its voltage. Label each one right after you
create it.
e. For each of the three electrode arrangements, perform the following steps:
• Draw in the equipotential curves for each electrode arrangement. Study the
spacing between your equipotential lines and, with your partners, identify regions
of strong and weak electric field. Explain your reasoning for each electrode
arrangement.
• The E-field lines, which go from the positive conductor to the negative conductor,
are always perpendicular to the equipotential lines. Draw in the E-field lines
using dashed lines. Clearly indicate the direction of the electric field at
several positions on each plot as indicated below. This will produce a map of
the E-field for your conductors.
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Make sure to draw the correct spacing (approximately) between the field lines.
The E-field is strong where the field lines are close together and weaker where
they are far apart. By looking at the shape of the E-field with the electrode
arrangement, how does the field lines pattern predicted by theory compare to
what you are actually observing? Explain your reasoning.
Repeat the mapping experiment for the three electrode arrangements.
The questions are setup to help you analysis each mapped field. It is
therefore important that all questions referring to each electrode
arrangement are answered.
3-2
Electrode Arrangements
Part 1: Parallel Plate
Before you start mapping out the field, take a few moments to predict how the electric
field and the equipotential curves should look like for this electrode arrangement.
a. Locate the equipotentials at voltages 3V, 4.5V 6V, 7.5V, 9V, 10.5V and 12V.
Make sure that the (i) fringe fields are also mapped out (i.e. the edge fields). As
well as (ii) behind the parallel plates as well.
b. Focusing on the middle of plate where the equipotentials are fairly uniform,
measure the distance ∆x between each of the following pair of potentials.
c. Calculate the E-field using E = −∆V/∆x between adjacent pairs of equipotentials
around the center of the parallel plates.
d. Compare the averaged E-field value with each individual value using a percent
difference. How do they compare? Record your data in a table.
Questions:
• Theory predicts that the electric field in between and around the center of the
parallel plates should be a constant (i.e. not considering the edges). Are the
values of the E-field at the points approximately constant within the experimental
uncertainty?
• Under what conditions will the field between the plates be a constant? Explain.
• What conclusions can you draw about the field strength at various locations? In other
words, identify regions of strong and weak fields on your mapped field.
• What is the field above and below the parallel plates? Should the field be zero?
Explain.
• Starting with your mapping results, is there symmetry in the parallel plate electric
field? Is there symmetry in the equipotential lines? Explain.
Part 2: Electric Dipole
Before you start mapping out the field, take a few moments to predict how the electric
field and the equipotential curves should look like for this electrode arrangement.
Questions:
• By looking at your equipotential curves, how does the electric field strength vary with
distance from one of the charge particles?
• What conclusions can you draw about the field strength at various locations? In other
words, identify regions of strong and weak fields on your mapped field and explicitly
show these on your plot.
• Compare the field map of the dipole with the mapped field of the parallel plates.
Account for the difference.
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Starting with your mapping results, is there symmetry in the electric field for the
dipole charges? Is there symmetry in the equipotential lines? Explain
Would it make any sense for field lines to cross? Explain your answer.
Part 3: Electric Dipole with Conducting and Nonconducting Circles
Map out at least 15 to 20 equipotential lines (say in 0.5-V increments) for this electrode
arrangement, especially around the insulator and the conductor. It will help you to “see”
what is going on physically. Also, be sure to map out the edges of the conductor and the
insulator.
Before you start mapping out the field, take a few moments to predict how the electric
field and the equipotential curves should look like for this electrode arrangement.
Questions:
• What are the potential and the electric field inside the conductor? Does your
measured value make sense? Explain.
• Why are the equipotential lines near the conductor surface parallel to the surface?
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What are the potential and the electric field inside the insulator? Does your
measured value make sense? Explain.
Why are the equipotential lines near the insulator surface perpendicular to the
surface?
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Challenge: why do the field lines “intensify” around the conductor?
What to turn-in:
• Each group generates only 3 mappings, one for each different electrode
configuration.
• Now make photocopies of the 3 mapping before drawing the equipotential lines so
that each group member has one copy of each electrode configuration.
• After a group discussion of the questions asked in the lab write-up, each group
member individually draws out the equipotential lines and electric fields, and answers
the questions.
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