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University of Puget Sound Introductory Physics Laboratory
4. Mapping electric potential
Name:____________________
Date:___________________
Objectives
1. To determine experimentally the lines of electric equipotential created
around conductors held at a constant voltage.
2. To create an electric-field map from a map of equipotentials.
3. To distinguish between electric field lines and lines of equipotential.
Equipment
Poorly conducting paper, conducting ink, DC power supply, digital
multimeter used as a voltmeter, conducting pins, light colored pencil, rubber
band, and cork board.
Helpful reading
Hecht Chapter 16 or Rex/Jackson Chapter 18 provides an introduction to
electric potential theory.
Background: Electric field
When buying groceries, we are often interested in the price per pound.
Knowing this, we can determine the price for a given amount of the item.
Analogously, it is convenient to know the electric force per unit charge at points
in space due to an electric charge configuration. Knowing the force per unit
charge, we can easily scale up to figure out the electric force. This electric force


per unit charge is called the electric field E . If a charge q feels an electric force F ,
 
then we say that the electric field at the position of the charge is E  F/q (N/C).
If the charge were changed to Q, the electric field would be the same but the

force on Q would be Q E . By convention, the electric field is determined by
using a positive test charge q. The direction of the electric field is simply the
direction of the force experienced by a positive test charge.
By determining the electric force on a test charge at various points in the
vicinity of a charge configuration, the electric field may be “mapped” or
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represented graphically by lines of force. The scientist Michael Faraday (1791 –
1867) introduced the concept of lines of force as an aid in visualizing the
magnitude and direction of an electric field.
The electric-field vectors
appropriate for a single, positive, point charge are sketched below.
The lengths (magnitudes) of the vectors are smaller at greater distances from the
charge. Why?
By drawing lines through the points at the tails of the vectors, we form
lines of force. The lines of force give a graphical representation of the electric
field. The direction of the electric field at a particular location is tangent to the
line of force at that point. The closer together the lines of force in a given region,
the stronger the electric field in that region. A positive charge released on our
map would accelerate in the direction of a line of force indicated by the arrows.
The motion of the charge may be more complicated, that is, the velocity may not
be parallel to the force.
Electric potential
Mapping the electric field is easy to describe but difficult in practice: we
can't move a test charge around and measure the force on it. But we can easily
measure potential, or voltage, differences. To move a positive charge from point
A to point B requires work supplied by an external force to move the charge
against the electric field (force). The work W per unit charge q done is called the
potential difference:
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VBA  VB  VA 
W
.
q
If a charge is moved along a path perpendicular to the field lines, no work
is done (W = 0), since there is no force component along the path. Then along
such a path (dashed lines in the sketch above), V=VB – VA = W/q = 0, and VB =
VA. Hence, the potential is constant along paths perpendicular to the field lines.
Such paths are called equipotentials. A map of the equipotential contours is just
like a topographic map, with potential playing the role of altitude. The
equipotential map and the lines of force map contain the same physical
information. In the following exercises, you will construct both kinds of maps
for the same potential configuration and explore the relationship between the
two different descriptions.
Exercises
OK, here comes the fun part. On the black, poorly conducting paper,
carefully draw with conducting ink a rectangular boundary close to the edge of
the paper. Then draw two conducting dots (about 0.5-cm in diameter and 2 cm
apart) in a dipole configuration, symmetrically placed inside the rectangle. After
the ink has dried completely, mount your paper on a cork board with plastic
pins. For each conducting shape drawn on the paper, push a conducting (metal)
pin through the ink and into the cork. Make sure that the bottom of the pin is in
contact with the conducting ink. Connect with alligator clips one conducting dot
to the positive (+) terminal and the other dot to the negative (-) terminal of your
power supply. Connect the outer boundary to the ground terminal. When you
turn the power supply on, the voltages applied to the conducting paint creates an
electric field in the surrounding area.
alligator clip
+
gnd
-
pin
+
ink
paper
-
cork
top view
side view
On your power supply, set the plus terminal at 10 volts and the minus
terminal at -10 volts. Turn it on and check the voltages at the supply terminal
with your DMM (in DC voltage mode.) Make sure you see the same voltages at
the other end of the wires, attached to the conducting paint. Check that different
parts of a single dot are at the same potential. Are they? Should they be?
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The potential at any point, relative to ground, can be determined using the DC
voltmeter. Connect one probe to ground and place the other probe at a point of
interest on the paper to measure the potential difference. An equipotential curve
can be determined by going to a point, determining the potential there, marking
the point with the colored pencil, then moving (by trial and error) to nearby
points that have the same potential and marking them.
Find several points that have the same potential and then smoothly
connect the dots to construct an equipotential. In the region surrounding each
dot, indicate at least ten equipotential curves. Try to map out the basic features of
the whole pattern, not just the fine details in one corner of the pattern. Make
sure to label the values of each equipotential on each contour.
Carefully sketch your result in the box below.
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Do you recognize any regular behavior? Do the values make sense? Be sure to
examine the potential behavior as you approach a conducting boundary.
Experimental electric-field mapping
The equipotential contours you just determined are like a topographic map, with
voltage playing the part of altitude. Can you visualize a three dimensional
image of your map? Moving a charge around on one of the contours costs no
energy. But to move from one contour to another costs an energy equal to the
charge times the potential difference between the contours (just like mgh for the
gravity problem.) This energy difference comes from work done against the
electric force. Like the equipotential map, you can also visualize the electric force
field (which of course carries the same physical information as the equipotential
map.)
It's not so hard to trace out the magnitude and direction of the electric
field experimentally. Connect two probes side by side with a short rubber band,
so that the probes are about 1 cm apart. Put one probe down somewhere along
one of your equipotential contours (call it your stationary probe.) Using your
other probe, test the voltage difference in a circle around the stationary probe.
What is the direction of the greatest decrease? This is the direction of the electric
field at your stationary point (the field is the negative gradient of the potential.)
Draw an arrow for your electric field, and note its strength next to it (in
volts/meter). Longer arrows for bigger fields, shorter arrows for smaller fields.
Pick several other places on the same equipotential contour and indicate the
direction and strength of the electric field. Sketch what you find in the box
below. In general, what is the direction of the field along an equipotential
contour? How does the strength of the field vary as you traverse a contour?
How does the field strength relate to the distance between the contours?
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Repeat this process of producing a potential and electric-field map on another
piece of paper. Paint conducting shapes of your own design. Examples include
two concentric conducting circles, or a conducting dot near a conducting U
shape, or a pair of conducting parallel lines. Sketch the resulting equipotential
and electric-field map in the box below.
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Before you leave:
Show your instructor your maps.
Explain what an electric-field map tells you.
Explain how to determine where to draw the electric-field lines if you have an
equipotential map.
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