An Experiment on Equipotential Curves
RAJESH B. KHAPARDE
AND
H. C. PRADHAN
Homi Bhabha Centre for Science Education
Tata Institute of Fundamental Research
V. N. Purav Marg, Mankhurd
Mumbai 400088, India
(rajeshkhaparde@gmail.com)
Abstract
We present here an experiment suitable for training in experimental physics at the
undergraduate level, which illustrates a novel technique of plotting the equipotential curves
for various geometries of electrodes. Using this technique, we plot the equipotential curves
for the following geometries: (i) two parallel plate electrodes, (ii) one point electrode placed
on the angle bisector of an L-shape electrode and (iii) two concentric cylinders. We then
study the uniqueness theorem. The method of images is also introduced to the students. For
this, we first plot the equipotential curves for a point electrode placed in front of a plate. We
then introduce another point electrode symmetrically on the other side of the plate and after
removing the plate, plot the equipotential curves for the dipole geometry. These plots of
equipotential curves help the students to visualize the electric field lines for various
geometries of electrodes.
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Introduction
The space surrounding an electric charge has a property called the electric field, which
follows the superposition principle. The electric field exerts a force on other charges. The
potential at a point in the electric field is the work done in bringing a unit positive charge
from infinity to the given point. An equipotential curve is a curve on which the potential is
same everywhere. The displacement of a charge over such a curve would require no work.
Since any two points on the curve have the same potential, there is zero potential difference
along the curve. The electric field has no tangential component along the equipotential curve
and it is always perpendicular to the curve.
Thus, if we consider two point charges separated by some distance, then the potential is the
same for a series of points, which when joined, form the equipotential curves for the dipole
geometry. In three dimensions, the equipotential curves form the equipotential surfaces. For a
point charge, the equipotential surfaces are concentric spheres and for a uniform electric field
they are planes normal to the direction of the electric field.
In this experiment to plot the equipotential curves, an acrylic tray half-filled with an
electrolyte is used as an electrolytic tank. Metallic electrodes of desired shapes are kept in the
tank. An AC potential difference is maintained between the electrodes, which causes a
distribution of potential in the electrolyte. By measuring the potentials (using a digital
multimeter) at different points, one can identify the coordinates of equipotential points and
plot the equipotential curves on the graph sheet. By taking a combination of electrodes of
desired shapes, one can plot the equipotential curves for various geometries.
Objectives
A) To plot and study the nature of equipotential curves for two parallel plate electrodes.
B) To plot the equipotential curves for a configuration consisting of a point electrode placed
on the angle bisector of an L-shape electrode.
C) To plot the equipotential curves for a geometry consisting of two concentric cylinders and
study the uniqueness theorem.
D) To study the method of images by plotting the equipotential curves for the plate and point
geometry, and the dipole geometry.
Apparatus
An acrylic tray, an AC power supply with a voltage stabilizer, a digital multimeter, a set of
metallic electrodes, a small amount of potassium chloride (KCl), a bucket with water, a set of
connecting cords, two measuring scales, two variable resistors and a spirit level.
Experimental Setup
The experimental setup (Figure 1) consists of a specially fabricated acrylic tray having
dimensions 50 cm × 75 cm × 5 cm. This acrylic tray has a thick acrylic base plate on which a
large sheet of laminated graph paper is pasted. X and Y-axes are drawn on this graph paper,
along with an appropriate scale and markings to locate the coordinates of a given point. The
tray can be leveled using the acrylic bushes fixed at the bottom. An electrolytic solution is
prepared using potassium chloride (KCl) and water. We use KCl since the mobility of K+
ions is nearly the same as that of Cl− ions. The tray is half-filled with the electrolyte. The
height of the electrolyte should be the same throughout the tray. Note that the electrolyte is
used to provide a low resistance path for the current, when metallic electrodes are placed in
the electrolytic tank and an AC potential difference is applied between them.
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Figure 1: Photograph of the experimental setup
A set of metallic electrodes is provided which include, flat plates, hollow cylindrical rings,
thin cylindrical solid rods, and an L-shape electrode. A thin rod is treated as a point electrode.
An AC (50 Hz) power supply, which has number of fixed voltage output terminals (in a step
of 1.5 V) from 0.0 V to 12.0 V, is provided. A 230 V AC voltage stabilizer is used to
maintain a constant input voltage to the AC power supply. The power supply is used to
establish the necessary potential difference between the electrodes. A digital multimeter with
long, pointed probes is provided to measure the potential at a given point in the electrolytic
tank with respect to a reference potential (0.0 V) of the power supply. One needs to insert and
hold the probe exactly vertical in the electrolyte. Two variable resistors (10-turn
potentiometers) are provided, which may be used in the potential divider mode to adjust the
potential applied to an electrode. A set of connecting cords, two measuring scales (30.0 cm
and 60.0 cm) and a spirit level are also provided.
Theory (For Part C)
In case of two concentric cylinders as electrodes (which form the boundary), the potential
distribution corresponds to the solution of Laplace’s equation (in two dimensions) in a given
region with the boundary conditions. The potential (rms voltage) at a point is a logarithmic
function of the distance of the point from the center of the cylinders. The graph of potential V
versus the distance r will be logarithmic in nature.
 r 
(1)
V = a ln   + b
 ro 
Where, V is the potential at a given point, ro is the radius of the inner cylinder, r is the
distance of the given point from the center of the cylinders, a and b are constants, which
depends on the charge distribution in the given region. The constants a and b can be
determined as follows:
Case 1: When r = ro
V = Vo ⇒ Vo = b
Case 2: When r = r1 ≠ ro,
We can write,
V = V1
r
V1 = a ln  1
 ro

 + Vo

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Hence,
a=
V1 − Vo
r
ln  1
 ro



Using Equation 1, we can write,

 r
 (V1 − Vo )ln 
 ro
V =

r 
ln  1 

 ro 


 
 +V
 o


(2)
Uniqueness Theorem
In a simplified form, the uniqueness theorem states that, the potential at any point between
the closed boundary surfaces (in this case, the electrodes) is completely and uniquely
determined by the Lapalce equation, given the values of the potentials on the boundary
surfaces, and is independent of the potential outside the boundary surface (i.e., electrode).
Procedural Instructions
Part A
Question 1: Why an electrolyte is used in this method of plotting the equipotential curves ?
Question 2: Why AC is preferred to DC in this method of plotting equipotential curves ?
Place two plate electrodes (exactly parallel to each other) in the electrolytic tank. The
distance between them should be less than their length. Apply a known potential difference
between the two plate electrodes. Use the given multimeter to measure the potential
difference. Using the graph paper pasted at the bottom of the tray, locate and mark points (on
a separate graph sheet), which lie at the same potential with respect to a reference potential.
Joining all such points will give you an equipotential curve. Plot at least four equipotential
curves for this geometry of electrodes.
Question 3: How would the equipotential curves for two infinitely long plate electrodes
look ? How do above plotted curves differ from those for infinitely long plate electrodes ?
Part B
Now take an L-shape electrode which has the same length for both the legs of the L-shape.
Place this electrode in the electrolytic tank. Take a point electrode and place it on the angle
bisector of the L-shape electrode at an appropriate distance. Perform the necessary
measurements and plot at least five equipotential curves for this geometry of electrodes.
Part C
Take two cylindrical electrodes. You may take one to be a point electrode. Place the two
cylinders in the electrolytic tank, so that the vertical axes of the cylinders coincide. Plot at
least five equipotential curves covering the region between the two electrodes.
Take the necessary set of readings and plot a graph of the potential V (at a given point with
respect to the reference potential) versus ln r, where r is the distance of the given point from
the axis of the cylinders and ln r is the natural logarithm (often pronounced as ‘lawn’) of r.
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Question 4: In the above case what change do you expect in the shapes of equipotential
curves when more electrodes (which are connected to higher potential) are introduced in the
electrolytic tank outside the larger cylinder?
Now introduce one or two point electrodes outside the larger cylinder. Apply appropriate
potentials (with respect to the reference) to these electrodes. Check whether the equipotential
curves plotted earlier, between the two concentric cylinders, are affected by the introduction
of these new electrodes. Try to understand your observations on the basis of the uniqueness
theorem.
Part D
Keep a plate electrode and a point electrode in the electrolytic tank separated by a distance of
9.0 cm. The point electrode should be kept exactly on the perpendicular bisector of the plate
electrode. Apply a potential difference (below 4.0 V) between the electrodes keeping the
point electrode at the zero potential point of the AC power supply. Plot at least five
equipotential curves for this configuration of the electrodes.
Now, place another identical point electrode symmetrically on the other side of the plate
electrode. Remove the plate electrode and apply to the second electrode, a potential exactly
twice the potential that was applied to the plate electrode. You may use variable resistors to
adjust the potential difference to the required value. Now, check whether the equipotential
curves plotted earlier remain the same or change. Try to understand your observations,
keeping in mind the method of images. Note that the point electrode introduced later is
equivalent to an image of the point electrode that was placed in front of the plate electrode.
Observations and Results
Part A
Answer to Question 1: We are interested in obtaining the distribution of potential in the
region (planer) between the two electrodes maintained at different potentials. A conducting
medium like electrolytic solution that provides a low resistance path for this alternating
current is necessary. The electrolyte allows us to measure the potential at any point in the
region between the electrodes using a digital multimeter.
Answer to Question 2: In this method AC is preferred to DC because DC will disassociate
the electrolyte, which will result in polarization of (charged ions in) the electrolyte. This
polarization will change the potential distribution in the electrolyte, which should essentially
be maintained the same throughout the measurement. Note that, at any instant the alternating
potential difference simultaneously changes at all points. Hence the rms voltage at any point
follows the same distribution that a corresponding DC between the electrodes would give.
Two plate electrodes were arranged in the electrolytic tank, parallel to and on either side of
the X-axis, at a distance of 8.0 cm from the X-axis and the equipotential curves for this
parallel plate geometry were plotted as shown in Figure 2.
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Figure 2: Equipotential curves for two parallel plate electrodes
Answer to Question 3: The equipotential curves for two infinitely long plate electrodes will
be straight lines parallel to the plate electrodes. The equipotential curves for plate electrodes
of finite length as plotted above curve near the end of the plate electrodes.
Part B
An L-shape electrode with length of 14.0 cm for both the arms was placed in the electrolytic
tank with its vertex at the origin and the two faces aligned along the X and Y-axes. A point
electrode was placed at a distance of 9.0 cm with coordinates (9.0, 9.0) from both the axes on
the angle bisector of the L-shape electrode and the equipotential curves were obtained as
shown in Figure 3.
Figure 3: Equipotential curves for a point and the L-shape electrode
Part C
A cylindrical electrode of radius 9.80 cm was placed in the electrolytic tank with its centre at
the origin. A point electrode was placed at the origin. We obtained the equipotential curves
for this geometry as shown in Figure 4.
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Figure 4: Equipotential curves for a point and a cylindrical electrode
In this part a separate set of readings was recorded to study the distribution of potential in the
region between the two electrodes. Here, the potential V at a given point and its distance r
from the origin were noted. From these data, a graph of potential V versus ln r was plotted as
shown in Figure 5. This graph was a straight line as expected from the theory.
Figure 5: Graph of potential V versus ln r
Answer to Question 4: The equipotential curves within the cylindrical electrode should
remain unchanged on the introduction of point electrodes (with any potential) outside the
cylindrical electrode. This is in accordance with the uniqueness theorem which states that
only the boundary conditions will determine the nature of the equipotential curves and the
electric field inside, irrespective of the charge distribution outside the boundary. In this case
the boundary conditions, i.e., the cylindrical electrode maintained at a given potential, remain
unchanged.
Now, without disturbing the earlier arrangement we introduced a point electrode outside the
cylindrical electrode and connected it to 7.54 V. We checked the equipotential curves
obtained earlier for any change. We noted that there was no change in the position and shape
of each of the equipotential curve plotted earlier. This experimentally verified the uniqueness
theorem. This also demonstrated the shielding effect of the conductors.
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Part D
A plate electrode of length 30.0 cm was placed on the X-axis, symmetrically with respect to
the Y-axis. A point electrode was placed at a distance of 9.0 cm from the origin on the
positive Y-axis with coordinates (0.0, 9.0). We plotted the equipotential curves for this
geometry as shown in Figure 6.
Figure 6: Equipotential curves for a point and a plate electrode
Now without disturbing the earlier arrangement we introduced another point electrode
symmetrically on the other side of the plate electrode. This point electrode was placed at a
distance of 9.0 cm from the origin on the negative Y-axis with coordinates (0.0, − 9.0). We
then removed the plate electrode and applied exactly double the potential difference between
the two point electrodes.
We plotted the equipotential curves for this dipole geometry as shown in Figure 7. As
expected from the method of images we found that equipotential curves above the X-axis
remained the same when checked with respect to the shape and the position of the earlier set.
Figure 7: Equipotential curves for two point electrodes (dipole geometry)
Analysis and Learning Outcomes of the Experiment
This experiment helps students to develop a link between a highly abstract theory and a
simple experiment. Here students are expected to compare the experimentally observed
equipotential curves with the theoretically predicted curves. It also gives an idea as to how
the equipotential curves would look in the real. The power of this method is in arriving at the
equipotential curves for an arbitrary electrode configuration for which a simple analytic
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solution of Laplace’s equation cannot be easily obtained. The deep mathematical foundations
of this experiment give the student an opportunity to see that mathematical theories which are
apparently quite abstract can have physical manifestations and these can be experimentally
observed.
This experiment emphasizes the development of conceptual understanding more than the
development of procedural understanding and experimental skills. Students are given
instructions on the use of the instruments like a digital multimeter and the AC power supply.
In this experiment, four questions have been asked with an aim to develop conceptual and
procedural understanding by predicting the variation in the experimental arrangement and the
equipotential curves. This experiment involves exploratory, inquiry, and illustration type of
experimental activities.
We describe below important aspects of conceptual understanding, procedural understanding
and experimental skills, which we expect students to get introduced to through this
experiment.
Conceptual Understanding
Electric charges, electric field, electric potential, potential distribution, electrolysis,
polarization of an electrolyte, equipotential surface or curve in an electrostatic field for
various geometries, Laplace’s equation and its solution, uniqueness theorem, method of
images.
Procedural Understanding
1) Choosing the proper orientation of the electrodes with respect to the axes drawn on the
graph paper.
2) Deciding the number of points to be identified so as to obtain a reasonably smooth curve.
3) Selecting the proper distances between the electrodes.
4) Choosing the proper potential difference to be applied between the electrodes.
5) Choosing a suitable potential, range and interval of values of the potential for which the
equipotential curves should be plotted.
Experimental Skills
1) Leveling the acrylic tray and arranging the apparatus (skill of alignment and adjustment).
2) Preparing the electrolyte by mixing appropriate amount of KCl in water (skill of
adjustment).
3) Measuring the potential at a given point in the electrolytic tank using the long pointed
probe of the multimeter (skill of using an instrument).
4) Locating a point at the given potential and reading the coordinates of the point from a
graph sheet (skill of recording and measurement).
5) Drawing smooth equipotential curves on the graph paper (skill of drawing).
6) Arranging the electrodes and keeping them stable and undisturbed throughout the
measurements (skill of alignment and control).
7) Using an AC power supply (skill of using an instrument).
8) Connecting the electrodes to the AC power supply without disturbing the arrangement of
electrodes (skill of adjustment and control).
9) Using a variable resistor in the potential divider mode, along with an AC power supply to
adjust the potential to a required value (skill of using an instrument).
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Possible Modifications
We present below a list of possible modifications which may be tried out and incorporated in
this experiment.
1) One can use some other easily available solutes like sodium chloride (NaCl) i.e. common
salt instead of potassium chloride (KCl) to prepare the required electrolytic solution.
2) One can use a stabilized variable frequency signal generator as a replacement of an AC
source (50 Hz). One can thus use a high frequency AC.
3) One can use ‘conducting ink’ with the appropriate connecting arrangements to replace the
metallic electrodes. Conducting ink can be used for making electrodes of any shape.
4) One can also use a copper plated board, which is used in the fabrication of electronic
printed circuit boards (PCBs). Here, one can use a simple technique of making the
required shapes on the board by dissolving the unwanted copper in ferric chloride (FeCl3).
One needs to be careful during the dissolving process to obtain the required shape of
electrodes and conductivity of the region between the electrodes.
5) It is possible to use a ‘conducting paper’ instead of the liquid electrolyte. ‘Conducting
ink’ can be used to make a ‘conducting paper’. One may use a DC source instead of an
AC source.
6) We know that the electric field lines will always be perpendicular to the equipotential
curves at every point and will point in the direction of decreasing potential. Thus one can
sketch the electric field lines for various geometries of electrodes.
7) One can have a variety of shapes and arrangements of electrodes and draw equipotential
curves. One can try some other shapes (like semi circles, squares, triangles, etc.,) and
combinations. One can try more complex distribution and geometries with three or more
electrodes.
8) One can use an L-shape electrode and point electrodes to plot equipotential curves and
study the method of images with the L-shape electrode as ‘reflector’.
Acknowledgements
We are thankful to Prof. Arvind Kumar and Prof. Vijay A. Singh for providing the necessary
support and facilities for the development and study under the Physics Olympiad and the
National Initiative on Undergraduate Science (NIUS) programme.
References
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Ltd., New Delhi, 1989, pp. 116-126.
2) Edward C. Jordan, Keith G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd
Ed, Prentice-Hall of India Pvt. Ltd., New Delhi, 1990, pp. 40-43, 67-68.
3) William H. Hyat Jr., Engineering Electromagnetics, 5 th Ed, Tata Mc-Graw Hill Pub. Co.
Ltd., New Delhi, 1995, pp. 129-131.
4) Frank G. Karioris, Kenneth S. Mendelson, Some Geometrical Features of Fields, Am. J.
Phys, 44 (8), 1976, pp. 749-753.
5) Hiroshi Murata and Mitsuru Sakuraoka, Electrostatic Potential on a Laboratory
Measurement Experiment, Am. J. Phys, 48 (9), 1980, pp. 763-766.
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Am. J. Phys, 36 (5), 1968, pp. 412-417.
7) Akio Saitoh, Folding Three-dimensional Model of Equipotential Curves, Physics
Education (UK), 24 (4), 1989, pp. 241-242.
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