Electric potentials and fields
In this laboratory you will explore the connection between electric field
lines and equipotential surfaces. Objects with
net electric charge attract or repel each other. If you want to change the
position of a charged object relative other charged objects, you, in general,
have to do (positive or negative) work. But sometimes it is possible to
move a charged object relative to other charged objects along a surface without
doing any work. The potential energy of the charged object does not change
as you move it. If an electric charge can travel along a surface without
the electric field doing any positive or negative work, then the surface is
called an equipotential surface.
Open a Microsoft Word document to keep a log of your experimental procedures
and results. Complete all the tasks (in blue). Answer all questions.
Activity 1
The concept of work
(a) The work W done on an object by a
constant force is defined as W = F∙d. It is equal to the magnitude
of the force, multiplied by the distance the object moves in the direction of
the force.
The SI unit of work is Nm = J.
An object travels from point A to point B while two constant forces of equal
magnitude are exerted on it, as shown in the figure on the right.
 Is the work done on the object by F_{1} positive,
negative, or zero?
 Is the work done on the object by F_{2} positive,
negative, or zero?
 Is the net work done on the object positive, negative, or zero?
(b) An object travels
from point A to point B while two constant forces of
unequal magnitude are exerted on it, as shown in the figure on the right.
 Is the work done on the object by F_{3} positive,
negative, or zero?
 Is the work done on the object by F_{4} positive,
negative, or zero?
 Is the net work done on the object positive, negative, or zero?
Work and the electric field
In the diagram on the right the red dot
denotes a positive point charge. Points W, X, Y, and Z and the point
charge lie in the same plane. Points W and Y are equidistant from the
charge, as are points Y and Z.
Draw the electric field vectors at points W, X, Y, and Z.
(c) A particle with charge +q_{e} travels along a straight line from
point W to point X.
 Is the work done by the electric field on the particle positive,
negative, or zero?
Explain using a sketch that shows the electric force on the particle and the
displacement of the particle.
(d) A particle travels from point X to point Z along the circular arc shown.
 Is the work done by the electric field on the particle positive,
negative, or zero? Explain!
Hint: Sketch the direction of the force on the particle and the
direction of the displacement for several short intervals during the motion.

Compare
the work done by the electric field when the particle travels from point W to
point X to that done when the particle travels from point W to point Z along the
path shown on the right.
Electric potential difference
A potential energy function is a function of the
position of an object. It can only be defined for
conservative forces. A force is
conservative if the work it does on an object depends only on the initial and
final position of the object and not on the path.
(e)
Suppose the charge in part (c)
increases from +q_{e} to +1.7 q_{e}.
 Is the work done by the electric field as the particle travels from from W to
X greater than, less than, or equal to the work done by the electric field on
the original particle. Explain!
 How is the quantity "the work divided by the charge" affected by this
change?
(f) The electric potential difference ∆V_{WX} between two points W and X
is defined to be the negative of the work done by the electric field on a charge q, divided by q, as q travels from W to X.
 Does this quantity depend on the magnitude of the charge that is used?
Explain!
 Does this quantity depend on the sign of the charge that is used? Explain!
When a net force does work on an object, its
kinetic energy changes.
W_{net} = ½m(v_{f}^{2 } v_{i}^{2})
= ∆K.
(g) A particle of charge q_{e} = 2*10^{6} C and mass m = 3*10^{8} kg
is released from rest at point W. The speed of the particle is measured to be
40 m/s as it passes point X.
 Is q_{e} positive or negative? Explain!
 What is the change in the kinetic energy of the particle as it travels from
point W to point X?
 Find the work done on the particle by the electric field between points W and
X.
 What is the electric potential difference ∆V_{WX} = V_{W}  V_{X} between
points W and X?
(h) Assume you have a test charge at rest at a distance of
2 cm from the charge on the right. You want to move it.
What path could you choose, so you would not
have to do any work? What is the shape of the equipotential
surface? (Remember that in general you can move in three
dimensions.) Explain your reasoning.
(i)
Find some equipotential surfaces for the charge
configuration shown on the right, which consists of two charged metal
plates placed parallel to each other.
What is the shape of the equipotential surfaces? Remember you are trying to
decide how a test charge could move so that the electric field does no
work on it. Sketch your predictions and explain your reasoning.
(j) Find some equipotential surfaces for the electric
dipole charge configuration shown on the right.
Sketch your predictions and explain your reasoning.
Activity 2
You now will calculate the electric potential at grid points in the in the xy plane
due to two small, uniformlycharged spheres. You will fix the
position of the charged sphere with charge q_{2}
and vary the position of the other charged
sphere with charge q_{1}. The xy plane is divided into a 26 x
26 grid. The upper left corner of the grid corresponds to x = 12.5 m, y = 12.5 m, and the
lower right corner corresponds to x = 12.5 m, y = 12.5 m.
The charged spheres can be placed anywhere on the grid in the xy plane,
as well as above or below the xy plane. You will calculate the potential at each grid point and construct a
surface and a contour plot of the potential. The contour plots will
display the equipotential lines. The electric field is perpendicular to the equipotential lines, E = ∇V.
You will draw field lines indicating the direction
and relative magnitude of the electric field in the vicinity of the charged
spheres and calculate the magnitude of the electric field at selected points.
The potential at r = (x,y,z) outside a uniformly charged sphere centered at r' = (x', y', z') is
V(r) = kq/r  r' = kq/((x  x'^{2}) + (y  y'^{2}) + (z  z'^{2}))^{½}.
The constant k has a value of 9*10^{9}
in SI units. If we measure q in
units of nC = 10^{9 }C, then kq = 9 q Nm^{2}/C.
Procedure:
 Open a Microsoft Excel spreadsheet. (Download an already prepared spreadsheet
HERE.)
 Cells B1  AA1 contain the numbers 12.5  12.5 in increments of 1 and cells A2  A27 contain the numbers 12.5  (12.5).
Cells B2  AA27 are the grid points whose x and ycoordinates (in units of m) are listed in cells B2AA2 and cells A2A27, respectively.
 Row 31 holds the total charge on each sphere (in units of nC) and the x, y and zcoordinates (in
units of m) of the positions of the centers of the spheres. We start with
q_{1} = 0 and q_{2} = +1 nC at x_{2} = y_{2} = z_{2}
= 0. (If we let the x and ycoordinates always be integers. we can
avoid "divide by zero" errors, since the grid points have half integer x and ycoordinates.)
You will later be able to change x and y coordinates of q_{1} using scrollbars. Therefore cells B31 and C31 contain formulas to
copy scaled scrollbar values from the cells below.
 Now find the potential due to the two
charges at grid point B2.
Into cell B2 type (or copy)
=9*$A$31/SQRT((B$1$B$31)^2+($A2$C$31)^2+$D$31^2)
+9*$F$31/SQRT((B$1$G$31)^2+($A2$H$31)^2 +$I$31^2)
This is the sum of V(r) = kq/r  r'
= kq/((x  x'^{2}) + (y  y'^{2}) + (z  z'^{2}))^{½}
due to the two charges.
 Now copy cell B2 into the other cells of your grid. The grid consists of cells B2  AA27.
 Highlight your grid (cells B2  AA27) and choose to INSERT a chart.
Choose a surface chart of subtype 3D surface (under All Charts, Surface).
Place the chart on Sheet 2.
(Chart tools, Design, Move Chart)
You will see a twodimensional surface plot of the potential outside a
small, uniformly charged sphere.
Excel 2010:
Select the graph, and from the "Chart Tools" menu choose Format,
Vertical (Value) Axis, Axis Options.
Excel 2013 or later: Select the graph, click the + sign next to
the graph, Axes, More Options, Axis Options, Vertical Value Axis, three
bar (rightmost) icon.
Now choose maximum 10, minimum 10, and major units 1.
 Construct another surface chart of subtype contour. Place the chart on Sheet 2.
(Excel 2010: Select the graph, and
from the "Chart Tools" menu choose Format,
Vertical (Value) Axis, Axis Options.
Excel 2013 or later: Select the graph, click the + sign next to
the graph, Axes, More Options, Axis Options, Vertical Value Axis, three
bar (rightmost) icon.
Now choose maximum 10, minimum 10, and major units 1.)
 You now have a surface and a contour plot of the potential outside a
uniformly charged sphere placed at the origin. The
contour lines are equipotential lines.
They are spaced in 1 V intervals. (Select a chart
style with enough colors and grid lines.)
 Now insert a scrollbar . Click the developer tab.
(If this tab is not available, click the File tab, click Option,
and click Customize Ribbon. Under Customize the Ribbon and
under Main Tabs, select the Developer check box.
To add the Developer tab to Excel 2011 on a Mac,
select Excel from the menu. It is between the Apple logo and File in the
upper left hand corner of the menu. From the drop down menu select
Preferences. In the Sharing and Privacy section, select Ribbon. In the
middle of the Ribbon dialog box you will see a box listing Tab or Group
title. Scroll through this list and find Developer. Check the box and
click OK. For Excel 2016 on a Mac, go to Excel preferences, view, in
ribbon show, developer tab.)
Click Insert, Form Controls, and choose the
scrollbar, then click in an empty cell, where you want the scrollbar to appear on
sheet 2. Drag the scrollbar to the orientation and size you want.
 To set the properties for the scrollbar, rightclick
it and chose select Format Control. Choose minimum value 0, maximum
value 24, incremental change 1, cell link Sheet1!$B$32, and click ok.
 Label the scrollbar by typing "xposition"
into a cell next to the scrollbar. This scrollbar
will let you move the charge q_{2} along the xdirection. In a
cell next to your scrollbar label type =Sheet1!B31. This copies the
xposition of charge q_{1} onto sheet 2, so you can monitor it while
you move the charge.
 Repeat the procedure above and insert a second scrollbar.
Rightclick it and chose select Format Control. Choose minimum value
0, maximum value 24, incremental change 1, cell link Sheet1!$C$32, and click ok.
 Label the scrollbar by typing "yposition"
into a cell next to the scrollbar. This scrollbar
will let you move the charge q_{2} along the ydirection. In a
cell next to your scrollbar label type =Sheet1!C31. This copies
the yposition of charge q_{1} onto sheet 2, so you can monitor it
while you move the charge.
 Click Insert, Form Controls, and choose the
checkbox, then click in an empty cell, where you want the
checkbox to appear on sheet 2.
Right click it, select Format Control, checked, cell link Sheet1!$A$32.
By clicking the checkbox you can change the sign of q_{2}.
 Now go to sheet 1 and enter =A32*21 into cell
A31 to set q_{1} equal to 1 nC.
Charge q_{2} is fixed at the origin.
Both charges have the same magnitude. You can move the charge q_{1}
to different positions with the scrollbars.
You can change the sign of charge q_{1} with the checkbox.
Describe you graphs and how they change when you move q_{1}. What do they tell you about the potential outside
two uniformly
charged spheres?

Produce a contour plot for
q1 
x1 
y1 
z1 

q2 
x2 
y2 
z2 
1 
5 
5 
0 

1 
0 
0 
0 
and paste the plot into your word document.

Draw approximately 8 field lines onto the contour plot.
The field lines should accurately reflect the strength and direction of
the electric field. At each point, the field lines must be
perpendicular to the equipotential lines.
 Pick a point on your graph at which to calculate
the magnitude of the electric field. Let the point lie on a field
line. Label the point A using the textbox tool.
Estimate the magnitude of the electric field at that point by finding
the change in the potential ΔV over some
small distance d along the field line. Let d go along the field line
from one side of the chosen point to the other side. Calculate E =
ΔV/d. (HINT!)
Convert your log into a lab report.
Name:
Email address:
Laboratory 5 Report
 In one or two sentences state the goal of this lab.
 Insert your log with the requested graphs and the answers to the
questions in blue font.
Save your Word document (your name_lab5.docx), go to Canvas, Assignments, Lab
5, and submit your document.