LPC Physics
Fields and Equipotentials
Fields and Equipotentials
Purpose:
To gain an understanding of electric fields and their relationship to electric potentials.
Equipment:
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Conductive Paper
Cork Board
Conductive Ink
Pushpins
Patch Cords and Alligator Clips
DIGI 35A Power Supply
Theory:
A. Some Laws and Definitions
Coulomb discovered experimentally that the electric force between two point charges
(1) is along the straight line that connects them, (2) is attractive for unlike charges,
repulsive for like charges, (3) is proportional to the magnitude of each of the charges
Q1 and Q2, and (4) is inversely proportional to the square of the distance R between
the charges. The magnitude of this force can be written
F = const.
Q1Q2
R2
Eq. 1
Equation (1) is known as Coulomb’s law and is our starting point. The difficulty in
working directly with this law, however, become apparent when we realize that in a
small piece of wire, for example, there might be 1020 separate charges (electrons) all
acting on each other, each exerting a force on every other one given by Eq. (1). And
if this isn’t already impossible enough, remember that force is a vector and all those
forces acting on each charge point in different directions.
So, to avoid having to use Eq. (1) directly, it is useful to introduce two related
concepts: electric field and electric potential. These concepts often seem very
“unreal” or “unphysical”. The primary purpose of this lab is to make them seem a
little less “unreal”.
Electric Field: The electric field at a given point can be defined as the force per unit
charge that would act on an electric charge placed at that point. The total electric
r
force, F , on a body at a point in space is then proportional to the electric charge, q,
of the body.
r
r
r r
Eq. 2
F = qE or E = F
q
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r
The proportionality constant is called the electric field, E , and is a vector property of
r
the point in space rather than of the charged body. The force, F , in Eq. (2) is related
to F in Eq. (1) in the following way. Let q of Eq. (2) be the same as Q1 in Eq. (1).
Then, if all of the forces F acting on Q1 due to all the Q2’s are summed up vectorially,
r
we get the total or net force on Q1 which is the force F , of Eq. (2). The electric field
may then be mapped by holding a small, lightly charged body (called a test charge)
near the objects producing the field and noting the direction and magnitude of the
force on the test charge.
(This is exact only if the test charge q is small enough so that it will not itself affect
the sources of the field. Mathematically, this is guaranteed by defining the field as
r
the limit of the ratio, F , for infinitesimally small q. However, for any practical
q
experiment, the smaller the charge, the smaller the force, hence making experimental
measurement difficult. In any event, it should be understood that the electric field
exists whether or not a test charge is present.)
The electric field can be visualized by using a vector whose length is equal to the
r
magnitude of E , centered at the point in space. Figure 1 shows the field of two point
charges depicted this way.
(a)
A charge of +3
(b)
A charge of -1
Figure 1
Field Lines: Just as in the case of a gravitational field, an electric field may be
represented by lines of force (or field lines) which are lines that, at each point, are
tangent to the direction of the electric field at that point. As a physical example,
consider that if an electric field exists in a region containing free charged particles at
rest, these particles will begin to move in the direction of the force on them, i.e.,
along the lines of force. The motion of these particles will then constitute a current.
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Figure 2
Figure 3
The advantage of field lines or lines of force is that they give us a much easier way to
visually depict the variation of both the magnitude and direction of the field in space.
The standard convention is to let the density of the lines represent the field strength.
In two dimensions the density is the number of lines per unit length perpendicular to
the lines. In three dimensions, the density is the number of lines per unit area
perpendicular tot he lines. All lines are continuous and start on positively charged
objects and end on negatively charged objects. Arrowheads can be inserted on some
lines to indicate the direction. Figure 2 shows the total (vector sum) electric field of
the two point charges shown in Fig. 1 depicted as arrows and Fig. 3 shows the same
field depicted with field lines.
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Electric Potential – Voltage: Similar to the definition of the electric field (as force
per unit charge), the electric potential is defined as the electric potential energy per
unit charge. The electric potential is a function only of the point in space and is
independent of the test charge (just as in the case for the definition of electric field).
An important difference, however, is that the electric potential is a scalar
quantity, unlike the electric field which is a vector.
The unit of electric potential is the Volt = Joule per Coulomb (V = J/C). (The
Coulomb is an amount of charge equal to the charge of 6.24 × 1018 electrons.) As
with any potential energy, the position of zero potential is arbitrary, and the important
physical property is the potential difference between two points. We note that the
potential difference is what is measured by a voltmeter. A voltage drop of 3 V across
a resistor, for example, means that the potential energy of a Coulomb of charge
decreases by three Joules as the Coulomb of charge moves through the resistor.
Equipotential Surface: A set of points having the same potential or voltage is called
an equipotential. In three dimensions, this will be an equipotential surface, but in
many situations, such as this experiment on a two-dimensional sheet of paper, the
equipotentials will be lines. In many problems it is much easier to find the
equipotential surfaces than it is to find the electric field directly. We now turn to the
question of the relationship between the equipotential lines and electric field lines.
B. Relation Between Electric Potential and Field
The work done by the agent moving a charge q a distance ∆x in the presence of a
r
constant electric field E is
r r
r r
Eq. 3
W ≡ F ⋅ ∆x = − qE ⋅ ∆x = − qE∆x cos θ
where θ is the angle between the direction of the field and the direction of the
r
displacement ∆x . The electric potential energy of the charge q in the electric field
will be increased as it is moved through ∆x by exactly the work given in E. (4),
(change in electric potential energy) = − qE∆x cos θ
Eq. 4
If we divide Eq. (4) by the charge q, the left hand side is just the change in electric
potential or voltage, ∆V. So,
∆V = − E∆x cosθ
Eq. 5
is the change in voltage or potential when moving a distance ∆x at an angle θ with
respect to the field.
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C
B
θ
A
Figure 4
In Fig. 4, the change in voltage or potential in moving from point A to point B is
negative. So we would say that point B is at a lower potential than point A. If we
then moved from point B to point C, the change in potential would be zero, since θ is
90o and cos90o = 0, making the right hand side of Eq. (5) zero. Points B and C are
therefore on the same equipotential surface (or line, in two dimensions). Additional
points on this same equipotential would be found by continuing to move at right
angles to the field lines. We thus have the important result: EQUIPOTENTIALS
ARE PERPENDICULAR TO THE FIELD LINES.
To achieve the greatest change in ∆V as we move a distance ∆x, we should move
parallel to the electric field so that cosθ = 1. In this case, ∆V = − E∆x , or
∆V
∆x
with ∆x in the direction of E.
E=−
Eq. 6
Equation (6) gives us a way to find E if we know the equipotentials. If we move in
the direction of greatest increase of V, then E points in the negative of that direction
and has magnitude ∆V ∆x . The units of E will be volts divided by a distance.
Similarly, if we know the field, we can use Eq. (6) to find the equipotentials
∆V = − E∆x when we move a short distance ∆x along E.
C: Relation Between Field Lines and Equipotential Lines
In order to have a consistent visualization of the pattern of equipotential lines and
field lines, it is useful to use the convention that all equipotentials will be voltageseparated by the same ∆V. Then, the spatial separation ∆x is given from Eq. (6) as
∆x = − ∆V E . We see that the spatial separation is inversely proportional to the
magnitude of E (the stronger is E, the closer the spacing of the equipotentials).
Let us now summarize the relationship between the electric field and potential and
between electric field lines and equipotential surfaces or lines.
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1. ∆V = − E∆x cosθ
∆V
2. E = −
( E || ∆x in the direction which maximizes ∆V )
∆x
3. Equipotentials are perpendicular to field lines.
4. The electric field is strongest where the field lines are closest together
(proportional to the density of field lines).
5. The electric field is strongest where the equipotentials are closest
together (proportional to the density of equipotential lines in two
dimensions).
Because the electric field is proportional to the density of both the field lines and
equipotential lines in two dimensions, the relative spacing between either set of lines
will remain constant. This means that if in a small region surrounding a particular
point in space, the field lines are spaced twice as far apart as the equipotential lines.
Although it is not mandatory, it is often useful to choose the density of the two lets of
lines to be the same. This creates a pattern of “squares” whose sides might be curved,
but which meet at right angles. Figure 5 shows the two-dimensional pattern of field
lines and equipotential lines that result from two concentric circular conductors with a
difference of 10V between them.
We notice in Fig. 5 that the electric field lines meet the conductors at right angles.
This is because each conductor is itself an equipotential. Observe also that even
though the spacing of the equipotentials and field lines changes drastically, their
intersections always form curvilinear squares.
D: How to Find Equipotentials and Field Lines in Two Dimensions
If you have familiarity with voltmeters, current and resistance, then read the next four
paragraphs. Otherwise, skip to Experiment.
Defining the two dimensional problem: A voltmeter measures a difference in electric
potential in volts. So, if we had a truly ideal voltmeter (so that it had infinite
resistance and drew no current whatsoever), we could attach one lead wire to a
reference conductor and with the other lead, measure the voltage that existed at any
particular point in space. This assumes that the presence of the voltmeter itself does
not disturb the pre-existing electric field patterns. For two-dimensional problems,
both of these problems can be readily overcome.
The problem of finding the electric field due to a real three-dimensional set of
conductors reduces to a two dimensional problem if the intersections of the
conductors with every plane perpendicular to a straight reference line are identical.
For example, finding the electric field around two infinitely long parallel cylinders
reduces to a two-dimensional problem, since there is no variation of electric field (or
anything else) as we move in a direction parallel to the axes of the cylinders. We will
call such a set of conductors a two-dimensional array.
A similar problem – two-dimensional current flow: It often turns out that the solution
to one problem is the same as that to another problem, which at first might seem to be
completely different. Such is the case with our two-dimensional conductor array. A
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sheet of paper which has deposited on one side a coating of carbon particles (like the
carbon particles compressed together to make a carbon composition resistor) is a truly
two dimensional resistor. We can imagine that if we attached wires to opposite sides
and put this “resistor” in a circuit containing a voltage source, there would be a
current in the surface coating and a voltage across the paper, from side to side.
Now suppose we press a piece of metal down on the conducting paper. Because
the electrical conductivity of the metal is much higher than that of the conducting
paper, the region of the paper in contact with the metal is all at nearly the same
potential. The electrons move freely through the metal with essentially no voltage
drop. We call these constant potential regions electrodes. If we have two or more
electrodes attached by wires to a power supply to establish their potentials, then
currents exist in the conductive paper surrounding the electrodes. We can measure
the electric potential at various points on the conducting paper between the electrodes
with an ideal voltmeter. The lines connecting points that are at the same voltage are
the equipotentials corresponding to that particular electrode arrangement.
Now you might ask whether these equipotentials in our two-dimensional
conducting paper are the same as those for a real 3D problem with axial symmetry
discussed in paragraph D.1. It turns out that they are the same, provided that the
conductivity of the paper is spatially uniform and the metal electrodes make good
electrical contact with the conducting surface. The conducting paper confines the
currents to two dimensions, i.e., the plane of the paper. The actual 3D electric field
pattern is very complex, but the components of the field in the plane of the paper are
parallel to the currents at every point. And the current distribution is truly a 2D
problem. So, even though there are components of the electric field that are not in the
plane of the paper, those components in the plane are the same as would exist in a
truly 2D problem.
Experiment:
1. Choose one of the electrode configurations that have been drawn for you on a
sheet of conductive paper. Layer scratch paper and carbon paper underneath the
conductive paper. Affix all three layers to a piece of cork with thumbtacks
through the corners.
2. Push a “double sided” thumbtack into each of the electrodes. Try to make contact
between the electrode and the flat part of the tack.
3. Set up the equipment as shown in Figure 1. Your electrodes may look different,
and this is fine. It doesn’t matter which is positive or negative.
DIGI 35A
voltage probe
V
C
-
G +
Figure 5
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4. Set the DMM to measure volts, and turn on the power supply. Set the voltage
output to 10 V.
5. Use an actual voltage probe or a patch cord as probe to detect the voltage changes
across the conductive paper. First check that one of your electrodes is at
approximately 10 V, and the other is at 0 V. (You should know which one will be
which).
6. Now you will trace out equipotential lines across the paper. Use your probe to
find all points on the paper that are at 1 + 0.1 V. Mark each point by pressing just
hard enough with the probe to leave a carbon-spot on the scratch paper (you may
want to test some spots by the edges before completing the entire procedure).
7. Move across the paper in 1 V-intervals. The equipotentials will be easy to see if
you mark enough points along each one. Make sure you mark enough points...a
minimum of 10 for each voltage level, more near anything “interesting”, i.e. the
point or edge of an electrode.
8. Make sure to identify the positions of the electrodes, and remember which is at 10
V and which is at 0 V.
9. Remove all pins and set aside everything but the scratch paper. Connect the dots
for each potential with smooth curved lines.
10. Label the equipotentials, e.g., 6 V.
11. Sketch in the field lines with a pencil (so you can erase) with a density (spacing)
of lines such that the field lines and equipotentials form curvilinear squares. Put
a couple of arrowheads on each field line to show the direction of the field.
12. Show on your paper three places where the electric field has a magnitude of 1.0
V/cm (100 V/m). Label these points as E = 1 V/cm. Find and label the strongest
and weakest field locations. Determine and show the magnitude.
13. Indicate the electric charge distribution on the electrodes by placing “+” and “-“
symbols on the electrodes where the field lines originate and terminate.
14. Repeat the above procedure with at least one other electrode configuration.
Questions:
1. How do field lines intersect equipotentials? Why?
2. For the electrode configurations that you used, where specifically was the electric
field the strongest?
r
3. Where is the charge density on the electrodes the greatest (relate to E )?
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4. Now that you are an expert on equipotentials, make a careful sketch of the
equipotentials and field lines shown in figures 6 through 9. Use at least a half
sheet of paper for each. Make sure the intersections of field lines and
equipotentials form curvilinear squares.
5. If the power supply voltage were doubled, how would the field pattern change?
The equipotentials?
+ 10 V
0V
0V
+ 10 V
Figure 6. Parallel Plates.
0V
Figure 7. Thin Cylinder Surrounded by Circular
Tube.
+ 10 V
+ 10 V
+ 10 V
0V
Figure 8. Thin Cylinders of Opposite Charge.
Figure 9. Thin Cylinders of Like Charge. (Note
conductor around edge of paper)
6. In Figure 10, which of these two points, X or Y, is the electric field stronger?
Explain why. How much work is required to move a +Q charge from X to Y?
Explain.
7. In Figure 11, where is the field stronger – point W or point Z? Sketch on the
figure what direction a positive charge placed at point Z would move. Sketch the
direction in which a negative charge placed at point W would move.
X
W
Z
Y
4 volts
3 volts
3 volts
2 volts
2 volts
1 volt
Figure 10
Figure 11
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8. The gray charge distribution shown in Figure 12 generates an electric field
corresponding to the following equipotential surfaces. The potentials at points A
and B are VA = 3.0 V and VB = 1.0 V.
a. On this diagram, sketch some of the electric field lines resulting from the
charge distribution. Is the charge distribution positive or negative (yes,
you have enough information to tell)?
b. Where is the electric field strongest? Explain.
c. How much work would it take to move a Q = 0.50 C point charge along a
straight line from B to A?
d. Now consider a semi-circular path from B to A as in Figure 13. To move
the Q = 0.50 C charge along this path, would it take more work, less work,
or the same work, as compared to part (c)? Explain.
e. Which takes more work: Moving charge Q from point C to point A, or
moving it from point B to point A? Justify your answer.
B
A
B
C
A
C
Figure 12
Figure 13
♣
Results:
Write at least one paragraph describing the following:
• what you expected to learn about the lab (i.e. what was the reason for conducting
the experiment?)
• your results, and what you learned from them
• Think of at least one other experiment might you perform to verify these results
• Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in
this laboratory.
♣
This lab was adapted from Physics 9C Lab Manual, The Staff of the Physics Department, University of
California at Davis, 2000; and from Physics 7B Labs, University of California at Berkeley.
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Clean-Up:
Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. How you divide clean-up duties between lab members is up to you.
Clean-up involves:
• Completely dismantling the experimental setup
• Removing tape from anything you put tape on
• Drying-off any wet equipment
• Putting away equipment in proper boxes (if applicable)
• Returning equipment to proper cabinets, or to the cart at the front of the room
• Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
• Shutting down the computer
• Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
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