Resource Lesson
Electric Potential Energy: Point Charges
Printer Friendly Version
In the following diagram, the central charge equals 10 µC.
Surface A (r
_{A}
= 3 meters) the potential equals:
V
_{A}
= kQ/r = (9 x 10
^{9}
)(10 x 10
^{6}
)/3 = 3 x 10
^{4}
volts
Surface B (r
_{B}
= 1 meter) the potential equals:
V
_{B}
= kQ/r = (9 x 10
^{9}
)(10 x 10
^{6}
)/1 = 9 x 10
^{4}
volts
If a 2 nC charge were to be brought in from infinity and placed on surface A shown above, the amount of work done on the 2 nC charge would equal
W = qΔV =
2 x 10
^{9}
(ΔV)
2 x 10
^{9}
(V
_{A}
 V∞)
2 x 10
^{9}
(3 x 10
^{4}
 0)
6 x 10
^{5}
J
We say that the 2 nC charge has gained an
electric potential energy
of
EPE
_{A}
= 6 x 10
^{5}
J.
By definition,
the absolute potential at a position infinitely far from a point charge is defined to be zero
.
Similarly, the amount of work done on the 2 nC charge to bring it in from infinity and place it on surface B would equal
W = qΔV =
2 x 10
^{9}
(ΔV)
2 x 10
^{9}
(V
_{B}
 V∞)
2 x 10
^{9}
(9 x 10
^{4}
 0)
1
.
8 x 10
^{4}
J
We say that the 2 nC charge has gained an
electric potential energy
of
EPE
_{B}
= 1
.
8 x 10
^{4}
J.
The difference between the 2 nC's electric potential energy at B and its electric potential energy at A would represent the work required to move the 2 nC change from a position on surface A to a position on surface B.
W
_{done by external force}
= q(ΔV
_{abs}
) = ΔEPE
In essence,
positive voltage changes
mean that an
external agent must do work
to move a positive charge to a new position in the field, while
negative voltage changes
mean that the
field would be doing all of the work
to move a positive charge to its new position in the field.
By definition,
charges flow naturally from points of high potential to points of low potential
. That is, when free to move, a positive charge would instinctively flow from surface B (high potential) to surface A (lower potential). Because of this, work done by electric fields (that is, when the charge moves along a field line in the direction of the field) results in a charge LOSING electric potential energy  that is, the electrostatic force causes a charge to move to positions of lower potential and less electrical potential energy consequently gaining KE.
An analogy can be formed between equipotential surfaces and altitudes from the surface of the Earth. At any given altitude, an object with mass has a certain amount of gravitational potential energy, PE
_{g}
= mgh where h is measured from some arbitrarily set zero level (usually the base of the hill). An external agent must do work against the gravitational field whenever the object's height, altitude, is increased  this results in the object gaining PE
_{g}
. The gravitational field does work on the mass whenever the object's height, altitude, is decreased resulting in the object losing PE
_{g}
.
By comparing the aerial view with the side view, you can tell that when the surfaces are closer together on the left, it signifies that the altitude is changing more rapidly, that is, that the slope of the hill is steeper. But regardless of which side of the hill a person climbs, he will do the same amount of work and gain the same amount of PE
_{g}
= mgh. Remember that this is the definition of a conservative field.
If instead, the aerial view where to be considered to be a series of equipotential surfaces, then the electric field is stronger on the left side than on the right side since the same changes in voltage occur in a smaller distance on the left than on the right. However, regardless of which direction a charge is moved from one surface to another, the same amount of work is done, since the charge gains or loses the same amount of electrical potential energy,
Work
_{done by electric field}
=  q(ΔV
_{abs }
) = ΔKE
The fact that these changes are
path independent
signifies that an electric field is also a
conservative field
 that is, the only thing that counts is a comparison of the ending position to the initial position, not the path taken between the two points. Remember, whenever an electric field does work on a charged particle, the particle loses electric potential energy and gains kinetic energy. This is analogous to gravitational fields. When gravity does work on a mass, it loses potential energy and gains kinetic energy.
If a system contains more than one charge, then the EPE of the system is the sum or the EPE of each pair of charge.
EPE
_{sys}
= k Σq
_{i}
q
_{j}
/r
_{ij}
In the following collection of charges, if each charge lies at the corner of a square of side
s
, then what is the EPE
_{sys}
?
Note that:
r
_{12}
= r
_{23}
= r
_{34}
= r
_{14}
= s
r
_{13}
= r
_{24}
= s
There are 6 "pairs" in this diagram:
q
_{1}
q
_{2}
q
_{1}
q
_{4}
q
_{2}
q
_{4}
q
_{1}
q
_{3}
q
_{2}
q
_{3}
q
_{3}
q
_{4}
The total electric potential energy of this system would equal
= k Σq
_{i}
q
_{j}
/r
_{ij}
= k (q
_{1}
q
_{2}
/r
_{12}
+ q
_{1}
q
_{3}
/r
_{13}
+ q
_{1}
q
_{4}
/r
_{14}
+ q
_{2}
q
_{3}
/r
_{23}
+ q
_{2}
q
_{4}
/r
_{24}
+ q
_{3}
q
_{4}
/r
_{34}
)
= k (q
^{2}
/s + q
^{2}
/s
+ q
^{2}
/s + q
^{2}
/s + q
^{2}
/s + q
^{2}
/s
+ q
^{2}
/s)
= k (

q
^{2}
/s
+

q
^{2}
/s
+

q
^{2}
/s
+
q
^{2}
/s
+
q
^{2}
/s
+
q
^{2}
/s
+ q
^{2}
/s)
= k (q
^{2}
/s)
Refer to the following information for the next nine questions.
Use the following electric field map to discern if you understand the concepts discussed so far regarding electric fields and electric potentials.
Where is the electric field strongest? L, M, N, R, S, T, U (Support your choice)
Where is the electric field weakest? L, M, N, R, S, T, U (Support your choice)
What is the direction of the electric field at R?
How much work would it take an external agent to move a charge from R to N?
What does the negative sign mean in the previous answer?
Does it take more work to move a 2 µC charge from R to L and then to T compared to going directly to T?
How much EPE did the 2 µC charge have while it was at rest at position R?
What is the 2 µC charge’s EPE at point T?
How much work was required to move the 2 µC charge from R to T?
Related Documents
Lab:
Labs 
Aluminum Foil Parallel Plate Capacitors
Labs 
Electric Field Mapping
Labs 
Electric Field Mapping 2
Labs 
Mass of an Electron
Labs 
RC Time Constants
Resource Lesson:
RL 
A Comparison of RC and RL Circuits
RL 
Capacitors and Dielectrics
RL 
Continuous Charge Distributions: Charged Rods and Rings
RL 
Continuous Charge Distributions: Electric Potential
RL 
Coulomb's Law: Beyond the Fundamentals
RL 
Coulomb's Law: Suspended Spheres
RL 
Derivation of Bohr's Model for the Hydrogen Spectrum
RL 
Dielectrics: Beyond the Fundamentals
RL 
Electric Field Strength vs Electric Potential
RL 
Electric Fields: Parallel Plates
RL 
Electric Fields: Point Charges
RL 
Electric Potential: Point Charges
RL 
Electrostatics Fundamentals
RL 
Famous Experiments: Millikan's Oil Drop
RL 
Gauss' Law
RL 
LC Circuit
RL 
Parallel Plate Capacitors
RL 
Shells and Conductors
RL 
Spherical, Parallel Plate, and Cylindrical Capacitors
Review:
REV 
Drill: Electrostatics
REV 
Electrostatics Point Charges Review
Worksheet:
APP 
The Birthday Cake
APP 
The Electrostatic Induction
CP 
Coulomb's Law
CP 
Electric Potential
CP 
Electrostatics: Induction and Conduction
NT 
Electric Potential vs Electric Potential Energy
NT 
Electrostatic Attraction
NT 
Lightning
NT 
Photoelectric Effect
NT 
Potential
NT 
Van de Graaff
NT 
Water Stream
WS 
Capacitors  Connected/Disconnected Batteries
WS 
Charged Projectiles in Uniform Electric Fields
WS 
Combinations of Capacitors
WS 
Coulomb Force Extra Practice
WS 
Coulomb's Law: Some Practice with Proportions
WS 
Electric Field Drill: Point Charges
WS 
Electric Fields: Parallel Plates
WS 
Electric Potential Drill: Point Charges
WS 
Electrostatic Forces and Fields: Point Charges
WS 
Electrostatic Vocabulary
WS 
Parallel Reading  The Atom
WS 
Standard Model: Particles and Forces
TB 
Advanced Capacitors
TB 
Basic Capacitors
TB 
Electric Field Strength vs Electric Potential
PhysicsLAB
Copyright © 19972020
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton