Phys 232 – Lab 4 Ch 17
Electric Potential Difference
Materials: whiteboards & pens, computers with VPython, power supply & cables, multimeter, corkboard,
thumbtacks, individual probes and joined probes, carbon-paper with traces
Objectives
In this lab you will do the following:
Analytically relate charge distribution, electric field, and electric potential for a rod
Computationally map electric field and electric potential (equipotential lines) for pairs of
charges
Experimentally map electric field and electric potential (equipotential lines) for various
source configurations.
Theory
In chapter 17, you learn about the electric potential (a.k.a. Voltage), that closely relates to the
work or electric potential energy for moving a charge in the presence of other charges.
The definition of electric potential difference is
V
E d .
Two important properties of the electric potential are:
The electric potential decreases most rapidly in the direction that the electric field points.
Throughout a conductor, if no charge is moving, then the electric potential is the same.
Equipotential Lines. The electric field throughout a region of space is often visualized with
arrows representing the magnitude and direction of the field at various locations. The electric
potential throughout a region of space is often visualized with equipotential lines, curves along
which the electric potential is constant. These are analogous to the ‘equi-elevation’ (contour)
lines featured in a topographical map. As with topo maps, where these lines are closely packed,
the potential changes steeply; where they’re spaced out, the potential changes more gradually.
I.

Uniform
charge -Q
Analytical – Potential and Field of a Uniform Rod
On a whiteboard, work the following problem (17.P.66).
b
Length L>>a,b,c
c
C
B
The long rod shown has a length L, which is much greater
than any of the other lengths, and carries a uniform charge –
Q.
a. Along the path A to B to C, draw the electric field at
several locations. This will provide you visual clues as to
whether it’s varying, whether we’re headed with or
against the field, etc.

l ’s along the path.

f 
E
d
l
c. Add up (or integrate) V
along this path
i
a b. Draw representative
A
from A to C. (Don’t forget that Ch 16 gives an
approximation for the field of a very long rod.)
Phys 232 – Lab 4 Ch 17
II.
2
Electric Potential Difference
Computational – Potential and Field of a Charge Pair
You’ll complete and run a program to determine a pair of point charges’ electric potential and
electric field throughout a region of space and represent those with a color gradient and contour
(or equipotential) lines and vectors (for the electric field)
a. Equipotential Lines
An equipotential is the curve (or 3D surface) that connects all observation locations that have
the same electric potential.
The situation you’ll model is illustrated in the figure below: two point charges, q1 and q2, both
contribute to the electric potential at an observation location. (As the text discusses, the
potential “at” a point is really the potential difference between that point and some reference
location; in this case, the reference location is out at infinity.)
Given the familiar expression for the electric field
due to a point charge and given the general
relationship between field and potential (see the
theory section at the beginning of this lab), the
electric potential at an observation location due to
a point charge is
q1
q2

V1 (ro )
1
4
o

q1
where ro

ro 1
1

ro

r1 .
Written in terms of vector components, the
magnitude of this separation vector is

ro
1
xo
x1
2
yo
y1
2
zo
z1
2
.
Similar expressions can be written for the electric potential due to charge 2.




From WebAssign, download the python code “Potential and Field.py” (open a blank VPython
file from the desktop, open the “Potential and Field” file through WebAssign, copy the code,
and paste it into the VPython window.)
Note: This program contains a mix of functions that you should be getting familiar with (like
linspace) and functions that you needn’t worry about mastering (like meshgrid and
the plotting functions); the latter are useful for efficiently visualizing voltage and field, but
we won’t use again in this course.
In the Calculations section, complete the xobs = … line so xobs is a list of 100 xcoordinates for observation locations, running from -1.5 to 1.5 meters.
Do the same with the yobs=… line (these being the observation location’s y-coordinates.)
Complete the ro1 =…, V1 =…, ro2=…, V2=…, and V =… lines so they give the
magnitudes of the separation vectors between the charges and the observation location
(Xobs, Yobs), each charge’s contribution to the electric potential there, and the total
electric potential there. Recall that in Python, xn is written x**n. Square root in Python is
either sqrt( ) or ( )**0.5.
Note: The power of the meshgrid function that defined Xobs and Yobs is that your
single line that defines V in terms of these will actually return not just one value of electric
potential, but the list of equipotential values for all observation locations!
Phys 232 – Lab 4 Ch 17




Electric Potential Difference
3
To test that you’ve gotten everything right so far, temporarily add a print statement just
below the V=… line, run the program, and check in WebAssign the first three V values in the
list that’s printed.
Note: if you’re working on your own computer and Python complains that “pylab” is not
installed, you’ll need to move your code to one of the lab computers to continue working on
it.
The program is incomplete, so it will crash, but not before printing.
In the Display section at the end of the program, uncomment and complete the
LineVs=… line so LineVs is a list of 401 voltages running from -10000 to 10000V. This
will set the equipotential levels to be displayed.
Run the program. The levelsPlotV=… line should be responsible for creating a contour
plot displaying 401 equipotential curves running from -10,000V to 10,000V, though the
curves near the point charges will be so close to each other they blur together.
b. Color Gradient
Of course, the 401 curves are just discrete samplings of the smoothly-varying electric potential;
to better visualize the smooth variation,
 Just below the levelsPlotV=… line, add a line that defines ColorVs to be a list of 1001
voltages, from -500V to 500V for which color shades will represent voltage (you’re only
colorizing from -500V to 500V so all the color gradation doesn’t get devoted to the space
near the charges where the voltages vary steeply.)
 Remove the # from before colorPlotV=… to activate that line which is responsible for
creating the color-gradient plot.
 Run the program and enjoy.
c. Electric Field
According to the equation given at the beginning of this lab, electric field and potential are
intimately related – electric potential difference is the path integral of electric field or,
conversely, electric field is the negative gradient (3D slope) of the electric potential. To see the
relationship between electric potential and field,
 The program already has lines of code that determine the x and y-components of the
electric field at several locations. You’ll notice some redundancy: Ex, Ey, and Ex2 and
Ey2 are calculated using slightly different lists of observation locations (XobsE, YobsE
and XobsE2, YobsE2); this is to avoid calculating E too near the two charges (where the
field is huge)
 Remove the # from the beginning of the quiverPlotE=… line and the beginning of the
quiverPlotE2=… line. Together, they will display the electric field on the same figure as
the electric potential.
 Run the Program.
 Based on the plot that’s generated, in WebAssign complete these statements about the
relationship between field and potential:
The electric field points ______________.
The electric field’s magnitude is greatest where ________________.
Where the electric potential is zero, the electric field is _____________.
Make sure partner names are written in a comment at the beginning of the program, save it,
and upload it.
Phys 232 – Lab 4 Ch 17
Electric Potential Difference
4
d. Varying Source Charges – Electric Potential and Field
To really appreciate how the Electric Potential and Field are related to each other, vary the ratio
of the two source charges.
 Change q2 to equal q1 and run the program to see how the field and potential change (you
may want to change the limits in ColorVs=… to better visualize the electric potential.)
 Change q2 to be –2*q1 and run the program to see how the field and potential change
(you may want to change the limits in ColorVs=… to better visualize the electric
potential.)
 Feel free to make any other changes you’d like to explore.
III. Experimental – Potential and Field
Two “point charges”
Experimental Set up
Get a sheet of resistance paper (carbon impregnated) with electrodes in the shape of two
small dots drawn in silver conductive ink. Tack the corners of the paper down on a
corkboard.
For the first part of the experiment, attach the power supply and the multimeter as shown
in the diagram below. Two of the leads should be connected to the electrodes with pins.
Set the dial on the multimeter to “DC V” so that it will measure differences in electric
potential (also called voltage). Turn on the multimeter and the power supply.
DC Power Supply
COM -5 V
+5 V
Digital
Multimeter
DC
1000V COM
Probe
Experimental Procedure
A. Find Equipotential Lines
1. Carefully sketch the shape of the electrodes on a piece of white grid paper.
2. The shape of an equipotential line is found by moving the probe on the paper to find
enough points where the voltage is the same that you can be confident of the shape of
the smooth curve that connects those points. For a give equipotential curve, you should
Phys 232 – Lab 4 Ch 17
Electric Potential Difference
5
find at least four points, mark them on your white grid paper, and then smoothly
connect the dots.
Note: Since your hand is about as conductive as the paper is, your measurements would
be affected if you rested your hand on the paper while take measurements, so be
careful not to.
3. Start by finding the curve where the voltage is +3 V. Carefully draw and label the
equipotential line on the white grid paper, not on the black conducting paper.
4. Repeat for the voltages +4 V, +2 V, +1 V, 0 V, -1 V, -2 V, -3 V, -4 V.
This should look similar to the contour plot in your program when q2 = -q1.
B. Find Electric Field Vectors
Suppose the voltage is measured at two points A and B that are a short distance d apart.
Midway between the points, the size of the component of the electric field along the line
connecting the points is approximately E|| VB VA d . If you choose points A and B so
that the line connecting them is along the electric field, the potential difference will be
maximum and you will be measuring the size of the whole electric field.
1. Disconnect the multimeter from the power supply. Also, remove the probe from the
multimeter. Connect the multimeter to the pair of probes that are held at a fixed
distance apart. Place the red wire in the “V” socket and the black wire in the “COM”
socket. In this configuration, the meter will display the electric potential difference
between the probes. A positive reading will mean that the red probe is at a point with a
higher electric potential, which means the component of the electric field points toward
the black probe (Since electric field is the negative gradient of electric potential).
2. Measure the distance between the tips of the probes.
3. For the following four points, determine the direction and magnitude of the electric
field. You’ll record the magnitude in WebAssign and sketch electric field vectors on the
white grid paper to indicate the appropriate direction and their relative magnitudes.
a. Straddling the +1V curve along the line directly between the two point sources (and
don’t forget to sketch an arrow on your white grid paper)
b. Straddling the +2V curve where it passes below one of the point sources (be careful
to find the appropriate direction of the field here and don’t forget to sketch an
arrow on your white grid paper)
c. Straddling the +2V curve where it passes above the same point source (be careful to
find the appropriate direction of the field here and don’t forget to sketch an arrow
on your white grid paper)
d. Straddling the -3V curve along the line directly between the two point sources.
b
d
a
c
Phys 232 – Lab 4 Ch 17
Electric Potential Difference
6
When you draw arrows representing the electric field at these points, scale them so that
the largest is not too big to draw on the diagram with the equipotential lines.
Remember that the tail of a vector should be where the electric field was determined.
The directions and relative magnitudes of the electric field at these points should
qualitatively agree with what you’d seen in your program for q2=-q1.
Point and Line Electrodes
Replace your black conductive paper with a conductive page with one point and one line
electrode.
In order to test that good contacts are made with the electrodes, touch various points on
each of the electrodes with the tip of the probe. The voltage readings for all points on a
single electrode should be very nearly the same (within about 0.1 volts).
For this, repeat steps A and B. It’s alright if you can’t find the equipotential lines too close to the
small electrode.
Write your group members’ names on the two white grid sheets and turn them in.