Equipotentials and Field Lines
2. Equipotentials and Field Lines*
Objectives:
surfaces.
This work concerns electric fields, electric potential and equipotential curves or
The learning objectives are the following:
1. To be able to explain how the electric field relates to electric force acting on a
charged particle.
2. To be able to say what electric potential is, what its units are and how the potential
relates to the electric field and the electric potential energy.
3. To be able to explain what electric equipotentials are and how they can be mapped
out using a volt meter and a tank of weakly conducting fluid.
4. To be able to define an electric field lines, give their properties, and say how field
lines must be related to equipotentials.
5. Finally, to be able to calculate and also to measure the electric deflection of
electrons in a CRT as a function of acceleration voltage, deflection voltage, and
geometry of the tube.
(These concepts are all described in your textbook.)
Reading assignment: Before you come to the lab read sections on electric field, electric field
lines, electric potential, the section on visualizing potential, equipotential surfaces, relation
between equipotentials and electric field lines.
Knight, Jones and Field (162): 20.4 The Concept of Electric Field, 20.6 Conductors and Electric
Fields, 21.1 Electric Potential Energy and Electric Potential, 21.2 Sources of Electric Potential
21.5 Connecting Potential and Fields
Serway and Vuille (212): 15.5 Electric Field Lines, 15.6 Conductors in Electrostatic Equilibrium,
16.1 Potential Difference and Electric Potential , 16.2 Electric Potential and Potential Energy,
Point Charges, 16.3 Potentials and Charged Conductors, 16.4 Equipotential Surfaces
Serway and Jewett (252): 23.4The Electric Field, 23.5 Electric Field of a Continuous Charge
Distribution, 23.6 Electric Field Lines , 24.4 Conductors in Electrostatic Equilibrium, 25.1 Electric
Potential and Potential Difference, 25.2 Potential Difference in a Uniform Electric Field
Pre-Lab Exercises: Do the following exercises before you come to lab. Bring them with you so
your TA can check them off when you enter the lab. Later, you will hand them in with your report.
1. (For Phys 212 and 252) What is the definition of the electric field at the point with position
coordinates (x, y, z) ? (This involves a small test particle with positive charge q. Use proper
vector notation.) Be sure to define any symbols you introduce where appropriate.
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*© William A Schwalm 2012
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2. If a small particle with electric charge q = - 3.5 x 10 Coulombs is placed at the point with
coordinates (-4.0m, 2.1m, 2.2m) where the x, y and z components of the electric field vector are
Ex = -10.5 Newtons/Coulomb,
Ey = 10.5 Newtons/Coulomb,
Ez = - 8.0 Newtons/Coulomb,
what is the force acting on the particle?
3. How is the electric potential energy of a charged particle with charge q at a given point related
to the electric potential, or the voltage, at that point? Give a formula.
4. Equipotential surfaces are imaginary surfaces over which the electric potential in a given
problem is constant. Electric field lines are directed curves such that at each point in space the
curve is directed along the electric field. Explain how (geometrically) the electric field lines are
related to the equipotential surfaces.
5. The figure at the right shows a cross-sectional
view of a conducting metal fin (above) at 10,000
Volts and a flat metal ground plane at 0 Volts.
Sketch in your best estimate of how the field lines
and the equipotentials would look. Represent field
lines as lines with arrow heads on them to indicate
direction, and the equipotentials as dashed curves.
Draw at least five field lines and three
equipotentials.
Scenario: Your group is assigned to a project at Nanosystems Inc. This
corporation makes electron microscopes and other surface probes
involving electron optics, in which electrons rather than light are used to
form images. The general shape of a cylendrical electrostatic lens is as
shown at the right in a cur-away view. This is a prototype for the
objective lens for a new instrument in early stages of development.
Cut-away view
The design team needs to know as much as possible about the electric potential
inside the lens. They have computed an electric field map, but to be sure of the
calculation, it was decided that your team would do water-tank measurements on
a scale model of a portion of the lens, and actually draw some of the
equipotentials by experimental measurement.
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Equipotentials and Field Lines
The water-tank model will look like half of a cross section of the lens. (Be prepared to explain
this.) Therefore, what needs to be done is to make careful measurements of equipotentials
around the segment of the lens as represented by this model.
The water tank method makes use of the fact that, when a weak electric current flows through the
water, the voltage (which is the same as the potential) at a given point can be measured using a
high-impedence volt meter.
In-class questions for group response:
1. Inside a good conductor in equilibrium, such as aluminum with only small current flowing,
there is almost no electric field. This is because the electrons move around and cancel out the
field. What does this tell you about the electric potential inside or on the surface of a conductor?
(You may consider the following picture of a path between points on the surface of a conductor.
How much work is done against the electric force on a charge as you move it along this path?
What does it tell you?)
b
a
2. On your white board, draw a prediction for several of the equipotentials for a parallel plate
capacitor formed by two parallel conductors as shown. There are three equipotentials that you
can draw exactly. Suppose for instance that the right-hand bar is at +3 Volts and the other one is
at 0 Volts (grounded). Then which three equipotentials do you know right away? Draw them in
your picture and also include your estimates for at least two other equipotentials.
3. On the white board, make a sketch of the mock-up right-hand lens segment as shown above.
On the sketch, show your prediction for several of the equipotentials. (In the water tank,
everything is two dimensional, so the equipotentials are curves rather than surfaces.)
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Equipotentials and Field Lines
Exploring the Apparatus:
The experimental setup consists of the
following items, as shown in the photo:
913 inch Pyrex baking pan, digital
multimeter, connecting wires, two alligator
clips, one long probing wire (banana-probe,
aluminum electrodes (two ½ inch diameter
cylindrical and two ¾ x 6 x1/4 inch bar
shape) clear plastic sheet (not shown,
1
8
1824 inch), Two sheets of graph paper
and one piece of graph paper to place
under the pan.
Below the tank of water you can place a coordinate grid to use for making position
measurements. If you attach the common lead of the voltmeter to one of the conducting objects,
you can call that zero potential. Other potentials would be measured relative to this reference
point. Set up a simple configuration of conductors, such as the one shown, and practice mapping
out an equipotential curve until you get some idea of how things work. Develop a method
involving teamwork. For instance one person could use the probe and locate points of the
chosen potential and then report the coordinates. Another could record them. It is best to record
them directly as points on a piece of graph paper, which should be provided. Team members
should take turns each of doing these tasks.
Work Assignment: The team must produce a report describing the equipotential surfaces and
electric field lines inside the electrostatic lens. For a check, you should first prepare a similar
study of the field lines and equipotentials inside a parallel plate capacitor, making note of the socalled “fringe field.” In each case you should collect sufficient data and draw at least three
equipotentials (in addition to the conductor surfaces) and use those to draw at least four field
lines (in addition to the center line).
Measurement Plan: Based on the exploration, your team should decide on a plan for mapping
out the equipotentials of any given set-up. This should include a description of what each team
member is going to do during the measurement, what data will be recorded and then how the
data will be used. Be specific. Outline your plan on the white board and be prepared to report it
to the class. Make a copy of your plan here.
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Equipotentials and Field Lines
(Plan continued)
Implementation: Put your measurement plan into action. Take the necessary data for the
parallel plate capacitor, and then also for the lens segment mock-up. Record the data and
relevant observations here. Your team will produce graphical representations of equipotential
curves and field lines. These should be photocopied and shared among group members. Attach
them also to your final report.
Recap: Before you leave the laboratory, take time to review with your team members the
learning objectives in the box above. Be ready to comment on how the lab activities have related
to each of these objectives.
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