Physics Laboratory
Last update: 2003.5.20
Experiment 7. Electric Fields and Equipotential Lines
Caution for TA
Purpose of Experiment
The main cause of electricity phenomena such as static electricity, lightning is called a charge,
and the representative electricity phenomena caused by charges is the force(electric force)
between charges. The electric force between charges is proportional to the multiplication of the
two charges and inversely proportional to the square of the distance between the two charges,
which is called Coulomb’s law.
It is convenient to introduce the concept of electric fields to understand forces between charges
those are a long distance away. That is; when a charge exists, electric field is formed around the
charge. And if another charge is inside this electric filed, it is thought that the electric field
influence (act a force on it) the charge inside the electric field. Quantitatively, the electric field is
defined by the force acting on a unit charge of (+)1 Coulomb(C), therefore, the electric force is
proportional to the electric field and the proportionality constant is the amount of charges on
which the force acts.
When the electric force is acting on a charge, if does not hold by someone, the charge start to be
accelerated (Newton’s law of motion) and the kinetic energy also increases. Where does this
kinetic energy come from? The (electric) potential energy stored in the electric field is converted
to the kinetic energy of the charge. In other words, to bring the charge in the original position,
we need to work just the same as we need to work to lift an object from the ground. The
minimum work needed to move a unit charge of (+) 1 coulomb(C) between two points is called
the difference of electric potential or voltage difference. The case of minimum work is the
extreme case in which the kinetic energy of the charge is constant when it moves from one place
to the other. If you set a convenient point a standard position, by choosing the electric potential
of that point a convenient value (0) the electric potential of some point is determined by
difference between the point and the standard point.
What electric phenomena would occur at some point near a charge or charge distribution is
determined by the electric field formed by the charge(s) or the electric potential around the point.
In principle, the electric fields of any charge distribution can be obtained by Coulomb’s law and
the superposition principle of electric fields, ie, the property of the vector summation, however in
the case of a point charge or uniformly distributed symmetric charge distribution, there is a easy
way to calculate the electric fields, which is called Gauss’s law. But Gauss’s law does not
contains new physics but is an another expression of Coulomb’s law.
In this experiment, you are expected to put two electrodes on a conducting paper[a] and apply
voltage difference between the two electrodes, and find equipotential points to obtain
equipotential lines by connecting those points. Also examine the electric field by obtaining
potential around a point. Especially detect the point of the probe automatically by using tablet
digitizer connected with a computer. And by drawing the equipotential lines on the screen using
analogue digital converter(ADC) to read the potential of the probe, you can draw equipotential
lines simply by rubbing the front page of the conducting paper on the tablet with a probe. Also,
many applications are possible such as the calculation of electric fields at an arbitrary point,
verification of Gauss’s and Faraday’s law, storing and printing of data files, and extra data
analysis through other programs.
[a: It is not always nessesary to use a conducting plate to measure voltage difference. But in this
experiment, it is needed to use a conducting plate that can conduct currents(though few) to find
equipotential points by measuring the path of the same the voltages. Ie; the electric field
produced by the conducting plate on which static current flow is approximately regarded as that
of fixed charges. This measuring method has 2 dimensional properties so the general
equipotential surface is a line.
(a)
(b)
From E. Hecht and Halliday and Resnick’s general physics book etc;.
In fact, on the earth, a considerable voltage differences always exist for various reasons. The
most evident example would be lightning. Besides, sometimes you may have feeled the existence
of voltage differences. Have you ever had those experiences?
Outline of Experiment
• Confirm that equipotential points exist around the electrodes with voltage difference.
Understand that connecting equipotential points in a plane make a equipotential line.
• Investigate that the equipotential lines chages as the shape of electrodes changes.
How does equipotential lines depend on the shape of electrodes?
• Study the relation between the equipotential lines and electric field.
How can you obtain electric field from equipotential lines?
Have a experience to conduct experiment with the aid of a computer, and know the way to
analyze data by a computer. Think that what part of Gauss’s law of electric field, Coulomb’s law
of electric forces can be derived from the shape of the equipotential lines, and understand those
laws.
Experimental Methods
These equipments are prepared in the laboratory. (Parentheses mean the number of them.)
Conducting paper on which graphite is deposited (1)
Electrode holder (2)
Tablet(1)
A pen and a probe in tablet(1)
DC power supply(1)
Analogue digital converter (1)
Input terminal (big)
A computer
Electric wire with elegator clip(2)
30 cm ruler (1)
Paper clip (1)
Coin or other shape electrodes (2, prepared by each students)
3.5” disket (1, prepared by each students)
If you need more equipments, inquire to your teaching assistant or the experiment preparation
room (19-114), or prepare them for yourself. And you need to know tablet digitizer, analogue
digital converter, computer program(Field Touch), and static voltage DC power supply.
The special point of this experimental set up is that the experiment can be conducted almost
automatically with the aid of a computer tablet digitizer and an analogue digital converter. Here
the parts where the voltage and position of the probe is measured are automated and the part that
can move the probe is remained manual. This experimental set up is composed of a conducting
paper on which graphite is deposited(10), a DC power supply to supply a constant voltage
difference between the two electrodes(13), tablet(6), pen(7), probe(8), analogue digital
converter(2), input terminal (big) (5), a computer, and program for measurement(Field Touch).
Figure 1. Semiautomated equipotential line experiment equipments
Colloidal graphite is deposited on the conducting paper. You can put electrodes of necessary
shape and size. And by DC power supply(13) apply a constant voltage between the two
electrodes. Use computer use tablet digitizer(ACECAD Co., Acecatll 5*5 Graphic Tablet) for the
use of tablet. But as the probe should be able to read not only positions(x, y coordinates) of the
probe but also voltages on the conducting paper, connect the electric wire(9) that is connected
with the conducting paper to the ground terminal (5) of analogue digital converter to measure the
standard voltage to the ground of input terminal (5), and read the voltage from the probe of stylus
pen(7) by wiring it to the input terminal (5) of analogue digital converter.
The analogue digital converter(2) should be inter-computer analogue digital converter(ADClone
Co., ACL-711B Multi-Function Data Acquisition Card). The computer is 486, and use it to
monitor the result and input or output data. Tablet is connected to the series connecter of PC, and
analogue digital converter is connected with the extension slot of the PC.
To conduct experiment semiautomatically, the program Field Touch can be divided by voltage
mode and electric field mode, and additionally provides liberal painting mode. Also it has several
options and print-screen function.
Bellows are the recommended and standard experimental methods
1) Place an electric conducting paper with graphite deposited, and fix it with a tape. [This
step can be already done by previous experiment] Place two coins as electrodes, hold it
with a electrode holder to secure the electric contact and fix those electrodes.
2) Connect the electrode holder to the DC power supply. [In this time disregard the sign of
the electrodes]
3) Connect the paper clip of the electric wire that is connected with input terminal (big) to
the electric conducting paper. [Any place is possible but bottom part of the paper is
recommended, and the parallel position is center of the electrodes]
4) Put the voltage control lever to 0 (the end point of counter-clockwise direction) and turn
on the power switch. Rotate the voltage control lever clockwise direction to reach about
10 V.[Caution; rotate the current control lever with CV LED is on. Do not try to rotate too
much]
[video]
5) Turn on the computer and the monitor, run the program Field Touch. Use the default same
interval equipotential mode and with pushing the button of the body of the tablet, move
the tip of the pen on the conducting paper and look at monitor to check that the
equipotential lines are recorded. In this time, check that the brush size, shape, standard
voltage, voltage difference, the width of errors and the position of electrodes(coins) are
proper and find the proper values of those by changing those parameters.
[video]
6) Go to file menu – new in the computer and remove the contents of painting area and draw
new equipotential lines and save it. Make sure that draw every equipotential lines of same
interval and check that the shape is coincide with the expected one.
[video]
7) Examine how the shape of equipotential lines are changes as you change inter electrode
distance or size or shape of the electrodes. You may use other methods in the Option –
Color scheme menu.
8) Select Field Vector in Drawing Mode menu. Push a point of conducting paper with the tip
of the pen and check that X is presented on the screen.
9) Obtain electric field vectors at the points on the equipotential line obtained from 6).
Confirm the relation between equipotential lines and electric fields.
10) Each group conduct an additional experiment with their special method. In other words,
use a particular(but meaningful) shape electrodes or place another conductor or an
insulator in the conduction path, (ie, this means that you use a conductor with a hole. But
because it is not possible to pierce a hole in every group's conducting plate, ask your
instuctor to use the prepared one.) or confirm the relation between the equipotential lines
and electric fields and so on.
You can print out the experimental results, if needed, submit a disket. Are these results
consistent with those of expected by Coulomb' law? If these are, what conclusion can
you make? If not, discuss the reasons.
Background Theory
Electric field is a vector quantity. If an electric field of a point is E(V/m), this means, by
definition, that the magnitude of the force acting on the charge, when you bring a charge of +1
Coulomb(C) to the point, is E(N/C) and the direction of the force is that of E. Now, the volatage
difference( or electric potential difference) is defined by the work done to bring a +1 Coulomb
charge from position E1 to E2 with constant velocity. In order to move a charge with constant
velocity, it is needed to apply a force of same magnitude and apposite direction as that of electric
force acting on the charge. Therefore voltage difference is the work done by this force
r2→ →
△U = U2 - U1 = - ∫E.dr
r1
and is independent of the path from r1 to r2.
(1)
If the electric potential is the same(U1=U2) at the two points of nonzero electric field, this means
that electric field is perpendicular to the displacement of the two points. In other words,
equipotential lines(or surfaces) are formed when you connect points of same potential, and the
electric field at each point of this equipotential lines(surfaces) is perpendicular to the
equipotential line(or surfaces). Electric field vector of each point formed by charges(or
electrodes) are connected to form a electric force line. Therefore the electric force line and
equipotential lines(surfaces) cross in a perpendicular way.
(a) In the case of uniform electric fields, electric force line is a straight line of same interval, and
the equipotential line(surface) is a straight line(surface) perpendicular to the electric field. (b)
Electric force lines produced by a point charge are straight lines to the radial direction, and at the
dense part(ie, close to the charge), the magnitude of electric field is higher. Equipotential
lines(surfaces) are concentric circles(spherical surface) and notice that the interval between each
concentric circles are different. (c) electric force lines and equipotential lines in the case that the
two charges that have same magnitude of charge and different sign are a small distant
away.(called electric dipole)
Voltages of different equipotential lines are different and usually on the equipotential line graph
the line(surface) has the same electric potential.
Electric forces described by Coulomb between two charges are proportional to the multiplication
of the two charges and inversely proportional to square of the distance of the two charges. Like
this, the force that is inversely proportional to square of distance is called an inverse square force.
Important forces in nature such as gravitation, electric forces and magnetic forces have this
characteristic. In the case of inverse square force, Gauss's law is valid. Ie, for any closed surfaces,
if you know electric flux, the value is simply proportional to total charges surrounded by the
surface. To obtain the proportionality constant, you express it a formula, then the electric flux
passing through the surface S is
→ →
.
Φe = ∮E dS
S
(2)
here, ds is the infinitesimal surface element and ∮ means the closed(surface) integration.
According to the Coulomb's law, as the magnitude of electric field at a point that is a distance r
away from a charge q is
E = q/(4πεor2)
(3)
and the the electric field direct away from the(+) charge, if you imagine a closed surface of
radius R centered the charge(called as Gauss's surface), as the magnitude of electric field is the
same for all points of the surface and the direction is perpendicular to this surface, you may find
out
Φe = [q/(4πεoR2)][4πR2] = q/εo
(4)
The last term of formular (4) is valid for any Gauss's surfaces and this relation is called Gauss's
law. In other words, electric flux from any closed surface is the quantity of total charge(in this
case, air or vaccum) divided by εo(= 8.85 x 10-12C2/N.m2). Repeatedly speaking, to emphasise,
Gauss's law is the characteristic of inverse square law. Ie, if Gauss's law is valid, the interaction
is proportional to inverse square of the distance.
More Thinking
 Conductors and electric fields
Electric fields in a conductor that has free charges(electrons) is 0 in equilibrium. Therefore if you
charge a conductor, the added charges are distributed on the surface of the conductor, the surface
of the conductor becomes a equipotential surface. So like a drawing below, if you place a
conductor in the uniform electric field produced by pararell plates, the electric field and
equipotential lines(surfaces) change. What can you know from Gauss's law in this case?
In this experiment, you may use static current, the electric field inside a conductor is not 0. But if
the conductivity of the conductor is much higher than that of the conducting paper on which
graphite is deposited, you can approximately interprete it as the same case.
 equipotential lines between two concentric circles.
Now like the pictures below, place two round electrodes on a conducting paper, apply voltage
difference Vo to flow a constant current I.
For convenience, let's put the electric potential of the inner circle of radius a 0. As the electric
potential V at a point that is a distance r away from the center is the voltage relative to the inner
electrode, if the resistivity of the conducting paper is unform,
V/Vo =
r
∫dr/r
a
-----------b
∫dr/r
a
(A1)
is valid. Here b is the radius of the outer electrode.
Ie,
V = Vo ln(r/a)/ln(b/a)
(A2)
, and equipotential lines are concentric circles. Compare this with the potential of concentric
cylinder or concentric sphere of charge distribution of +q and -q in inside and outside cirles.
References
• Tablet digitizer
• Analogue digital converter
• Computer program(Field Touch)
• DC static power supply
• Analysis method of measured data
• Analysis method via graph
• Coulomb- A lucky person who found inverse square law
• Henry Cavendish-A rich natural scientist who showed the inverse square law is valid
• Carl Friedrich Gauss-A genius who counted numbers before he learn how to say.