Lab 3: Electric Fields II
Introduction
This week's lab has two parts. In the first part, you will use the EM Field
computer program to continue Investing the properties of electric fields. In the
second part, you will create, and measure the properties of a real twodimensional electric field.
Objectives
• To explore the electric field produced by a line of charges.
• To gain experience with the concept of electric flux.
• To verify Gauss' Law.
• To measure the properties of real two-dimensional electric field.
Equipment
• The EM Field computer program.
• Two dimensional field apparatus.
• DC Power Supply.
• Digital Multimeter (DMM).
Part I(a): Examining the field due to a line charge
A. What Is a line charge?
If you charge a conducting rod the charge will spread over the entire rod. If there are no other nearby
charges, the charge will spread uniformly. We can simulate such a charge distribution in EM Field by
placing point charges at equal distances along a line. Select File/Get charges or currents from file from
the menu bar, then double-click on qline.emf. Note that there is one (+1) charge at each grid point. The
line charge could be given as 1 charge per grid unit. In SI units this would be 1 C/m (one coulomb per
meter). A line charge is specified in term of linear charge density, charge per unit length. The symbol to
represent linear charge density is the Greek letter lambda (), where "l" suggests length.
1.
How could you change the charge density? You increase each charge unit to 2 units, and thus
have 2 C/m. To see a way of changing the density, open the charge file qline2.emf. What is the charge
density for this configuration? ______________________.
B. Electric field versus distance from a line charge
You know that the electric field from a point charge is inversely proportional to the square of the distance
from the charge. That is, E  1/r2. Use the field vectors to find out how the electric field depends on the
distance from the line charge. Complete the table on the next page.
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qline.emf
qline2.emf
Distance from line
(in grid units)
Length of field
arrow (in cm)
Distance from line
grid units)
(in
Length of field arrow
(in cm)
1
2
4
1.
Does E  1/r2? Use the program Graphical Analysis to determine the power law behavior
by fitting a curve to E versus r using the functional form y = AxB. Report the power,
B:__________.
2.
How does E depend on ?______________________.
EM field allows you to view a line charge from one end. Select Sources/2D charged rod from the
menu bar. Place a rod near the bottom of the screen. Drop a few field vectors to find out how the
electric field depends on the distance and direction from the rod. Are your results consistent with
the qline results from above?_________________________________________________.
Part I(b): Gauss’ Law
Gauss' Law is one of the most powerful, yet mystifying, statements in physics. Its mathematical
form is:
 
E
  dA  q /  o . In words, Gauss' law states that if you enclose a charge in a closed
mathematical surface, then the electric flux integrated over that surface is proportional to the
charge enclosed by that surface. The electric flux is defined as the product of the component of
the electric field perpendicular to the surface times the area of the surface.
What is the meaning of flux and Gauss' law? EM Field will help you get a feeling for the meaning
of both. For a brief explanation of how EM Field shows the flux through a surface, select
Option/How we display Gauss' law from the menu bar. Read the text, clicking on the screen for
each new graph.
A. Flux from a single charged rod
Place a 2D charge rod with charge density +3 in the middle of the screen. To draw a Gaussian
surface around the rod, select Field and Potential/Flux and Gauss' law from the menu bar. Now
use your mouse to draw a circle with a radius one or two grid units around the rod. When the
cursor is near the starting point you can release the button and let the program complete the circle
for you. Remember that this circle is actually the end view of a long cylinder enclosing the rod.
Note that the electric field vector was always (approximately) perpendicular to the surface The
gray bars outside the circle represent the flux though each strip of the cylinder. "Q=3" means that
the value of the charge enclosed in the cylinder is three units.
1.
Draw a second circle with a much larger radius around the rod. The program again says
"Q=3," but the electric field was much smaller at the surface. How can the total flux be te
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same through these two surfaces? (Hint: Consider the definition of flux)_______________
___________________________________________________________.
2.
Draw a second circle with a much larger radius around the rod. The program again says "Q=3,"
Select Sources/Add more sources from the menu bar. Place a rod with charge density -3 in the
center of the two circles and remove the +3 rod. What is different about the flux and how is it
represented by the program?____________________________________________________________
______________________________________________________________________.
3.
Move the -3 rod to the edge of the screen and place a +3 rod in the center of the circles again.
Select Display/Unconstrain so that you can place the rod anywhere on the screen. Now move it
near the surface, but still inside the inner circle. The field, and flux, near the charge is much
larger, but the total flux is still given by "Q=3." Explain how this can be: __________________
_________________________________________________________________________.
4.
Finally, move the rod outside the inner circle. The program says that the enclosed charge, and
thus the total flux, is now zero. Explain how that comes about:___________________________
_________________________________________________________________________.
B. Flux from two charged rods
Place both the +3 and -3 rods outside the smaller circle, but within the larger circle. Note that there is no
charge inside the smaller circle, while there are charges, but no net charge within the larger circle.
1.
Explain how the total flux through each of the two surfaces is zero, even though the charges
Inside are different. You may find it helpful to print your screen (select File/Print screen from
the menu bar) and point to the positive and negative regions of flux in order to make your
point._________________________________________________________________________
___________________________________________________________________________________.
2.
Make notes on the printed page showing where the field direction is away from the positive
charge and toward the negative charge, and the sign of the flux In each case.
Detach pages 2 and 3 from your lab manual and attached them to the worksheet for Part II.
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Part II: Measuring a 2-dimensional electric field.
Introduction
You were first introduced to the field concept with the gravitational field.
Gravity, you remember, is not a force that acts when two objects touch. It is a
force that seems to act over a distance. Physicists have never been happy with
the idea of “action-at-a-distance.” Rather, they prefer to think about a
gravitational field that is created by one object, like the sun. The field exists at
the location of the second object, like Earth, and acts on it there. Thus the force is
a local, not a distance force. In the same way, physicists have found it useful to
describe the electric interaction in terms of an electric field created by one charge
acting on a second charge at the location of that charge.
Defining the electric
field
In principle, a field can be measured by measuring the force on an object located
in that field. The force is then divided by the object’s charge (E = F/q) , in the
case of the electric field, or mass (g = F/m), in the case of the gravitational field.
The units of the electric field are newtons/coulomb.
Electric field lines
The electric field can be visualized by means of field lines. An electric field line
is a line which is at every point tangential to the electric field. That is, it always
points in the direction of the field and it traces out the path that a charge would
follow if it were allowed to move at constant speed in the direction of the electric
force on it.
Electric potential
difference
A direct measurement of a field is very difficult. We will use an indirect way of
measuring the electric field that will involve the measurement of electric
potential difference.
The electric potential difference is defined as the work done on a unit positive
charge in moving the charge from one point to another in an electric field. The
units of potential are work per charge or joules/coulomb. One joule/coulomb is
defined as a volt.
As is shown in the textbook, the electric field, in one dimension, is the derivative
of the potential. For a three-dimensional field, we must use the three
dimensional rate of change of the potential. Thus, in component form:
dV
dV
dV
Ex = – dx , Ey = – dy , Ez = – dz
(1)
We can see that another unit for electric field is potential per unit distance or
volts/meter. (You can check to see that V/m and N/C are equivalent units.)
Because E is related to a spatial derivative of V, we can use a difference method
of measuring E. For points a and b close together we approximate the derivative
by a difference equation:
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Vb – Va
dV
V
Ex = – dx = –
=– x –x
x
b
a
(2)
The potential difference, Vb – Va, can be measured with a potential-difference
meter, commonly called a voltmeter. The quantity xb – xa can be measured with
a ruler. From these two measurements we can calculate Ex. We can do similar
measurements for the other components, Ey and Ez.
The negative sign means that the electric field component along any axis points
opposite to the direction in which the potential increases. Thus, the electric field
vector at any point in space always points opposite to the direction of the greatest
rate of change of potential and has magnitude equal to this rate of change.
Equipotential lines
and surfaces
It is also useful to define the concept of equipotential lines and surfaces. A point
charge can be moved without doing any work along an equipotential line (in 2
dimensions) or along an equipotential surface (in 3 dimensions). If no work is
done then the potential must be the same everywhere. Clearly there must be
zero electric force in the direction of motion along an equipotential.
This is all you need to know about electric fields and potentials to begin the
experiment. You will discover some of their properties from your observations.
The Experiment
In this experiment, we will set up a two dimensional electric field in a thin sheet
of slightly conducting material (called teledeltos paper) by applying a potential
difference between two electrodes. Everything we have said so far is the same
for this configuration. The potential difference means there is an electric field,
and the force due to the field causes a flow of charge (current) in the sheet. We
can be sure that the direction of the electric field is parallel to the faces of the
sheet, of, it were not, charge would flow to the faces and build up until the field
became parallel. Potential differences are measured by a voltmeter. In this
experiment, we can use a type of meter that does not require very much current
to operate. By using this meter, we can be sure that the field patterns are not
distorted by the presence of the meter.
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Electric Fields II
1. Equipotential Lines
Procedure
• Place a sheet of the conducting teledeltos paper on the mapping board that
has two bolts through it. These two bolts provide two point-like electrodes.
• Connect the power supply, multimeter and potential probe as shown in
Figure 1. (The potential probe is the one with only one point.).
+
–
Power Supply
Probe
Volts
V
COM
Figure 1. Using the potential probe
• Now touch the probe gently to the left-hand electrode. Adjust the power
supply to read 12V on the multimeter.
• You are now ready to look for equipotential lines. Gently probe for a point
where the multimeter reads 10V. Mark this point with a pencil. Now find
enough other points at 10V to be able to draw a smooth equipotential line
connecting the points. You can probably make a closed curve.
• Go back and do the job again, but this time measuring the equipotential line
at the 8V potential. Make sure you measure enough points so that you can
draw a smooth curve through the points. Can you make a closed curve this
time?
• Repeat this procedure for equipotentials at 6, 4, and 2 volts. Each time, make
sure you determine enough points so that you have an accurate indication of
shape and spacing of the equipotential lines.
• Label the equipotential lines with their potentials.
An equipotential line was defined as the line along which a charge can be moved
without doing work. Charges can move freely along a conductor (much more
freely than on the teledeltos paper). If you were to place a good conductor along
such a line, then, even though charges would be free to move, there would be no
force available to move them. Therefore, the potentials at other locations on the
sheet would not be changed. Let’s check this.
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Electric Fields II
• Place the potential probe on the 8-volt line. Press the edge of a short piece
of a ruler covered with aluminum foil along the 6-volt line. Hold it down
with your fingers. Does the voltmeter show any change?
Procedure
• Rotate the ruler until it is at right angles to the 6-volt line and press it down.
Does the voltmeter show any change now? If it does, find one or two points
along the new 8-volt and 4-volt lines.
2. Field Lines
Method
Procedure
•
The field probe consists of two points spaced about 1 cm apart. The two
wires of the electric field probe will be connected to the DMM (one wire to
the black (com) terminal, one wire to the red (“V”) terminal). Connected in
this manner, the field probe measures the potential difference between the
two points.
•
You can find the direction of the field by first touching the black probe to the
paper as before. Next, without removing this probe, choose a location for
the second, gently touch the point to the paper and read the potential
difference. Rotate the probe, trying different directions until you get the
maximum potential difference.
•
The DMM reads the potential difference between two points one cm apart.
To find the electric field at the midpoint between the two points, divide the
voltage by 0.01 m.
• Connect the field probe to the meter as discussed above and illustrated
below.
+
–
Power Supply
Probe
Volts
V
COM
• Starting with one probe a few millimeters from the left-hand electrode, find
the direction of the maximum potential difference and make pencil marks at
both probe points. Record this maximum potential difference V.
• Move the probe so that the first point is where the second was, and repeat
this procedure. Repeat this all the way out to the second electrode.
• By connecting the dots, you can draw a field line. Place arrows on the field
line in the direction of the field.
• Follow one other field lines, starting from the left-hand electrode at a
different direction, as suggested below. One field line should be very nearly
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a straight line from one electrode to the other.
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Names
Section
Electric Fields II
Data Analysis
1
•
Choose the line that is closest to a straight line from one electrode to the
other. For that line you have a series of measurements of V and x , where x
is the distance from the left-hand electrode to the midpoint of the two points
marks of the electric field probe. Fill in the table with the values of V, and x
.
•
Calculate E for each point. Because x = 1 cm = 0.01 m, E is just V/0.01 m,
(see Equation 2).
•
Plot a graph of E versus x . (Review your graph plotting technique from the
Physics 150 Laboratory Manual if necessary.)
2
3
4
5
6
7
8
9
10
11
12
13
V (volts)
E (volts/m)
x (m)
Hand in your teledeltos paper and the electric field graph with these worksheets.
Questions
1.
Based on your graph, is the electric field constant between the two
electrodes? If not, is there a region in which it is very nearly constant?
Describe the region.
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Electric Fields II
2.
Where is the component of the electric field along your line strongest? Is this
component of the field zero anywhere? If so, where?
3.
Draw the field and potential lines between the point and straight-line
electrode shown below. Hint: you need only consider the data you have
taken and the symmetry of the electrodes.
0V
6V
4.
Explain, in your own words, why the field probe method gives you the
magnitude and direction of the field.
5.
On the basis of your observations, what angle is made between field lines
and equipotential lines?
6.
Define, in your own words, an equipotential line.
7.
Do equipotential lines ever cross each other? Explain. Hint: What would it
mean? Think of being a charged particle on an equipotential when you came
to a crossing.
8.
Can field lines ever cross each other? Why?
9.
Why don't charges move along an equipotential line? Hint: What causes a
force on a charge?
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10. Suppose a thin straight wire were placed between the two electrodes as
shown. Draw the resulting field and equipotential lines.
Wire
11. Suppose a thin straight wire were placed as shown below. Draw the field
and potential lines that would result. Remember, the wire is an equipotential
line.
Wire
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