TOPIC 2. ELECTRICITY
Topic 2A. Electric Charges and Forces
ELECTRIC CHARGE
Objects, including elementary particles like the electron and proton, have a number of
properties. The one we have studied the most so far is mass. The mass of an object is important
in two of the basic laws of mechanics. First, from Newton's second law of motion, the mass of
an object determines its response to a force: the acceleration +t that results from the application of
a force Jt to an object is related to its mass 7 by Jt œ 7+t. Second, the mass of the object is
involved in gravitational forces through Newton's law of gravitation: the magnitude of the
gravitational force between two objects of masses 71 and 72 separated by a distance < is
Jgrav œ K71 72 Î<2 .
Another property of objects is their electric charge. What leads us to think that there are
such things as "charges," anyway? Think about what you have learned about "forces." A force
can be thought of as a push or a pull that may act on an object. Usually, the force that acts on
some object "A" is caused by some other object which is in direct contact with object "A." For
instance, a hammer might strike a nail; then the hammer is exerting a force on the nail.
According to Newton's third law, the nail also exerts a force on the hammer (equal in magnitude,
opposite in direction). This type of force is often referred to as a "contact" force. There is
another type of force, in which direct contact between objects is not required. Suppose you hold
a book up in the air, and then let it fall. What made it fall? You know that a gravitational force
exerted by the earth on the book was responsible for the book's fall. (By Newton's third law,
there is also a gravitational force exerted by the book on the earth.) You have also observed
magnets appearing to exert forces on each other, even though they are not in direct contact.
These forces have been called "action-at-a-distance" forces.
Now consider some other phenomena you have encountered in the past. When you rub a
comb through your hair, you may have noticed the comb pick up small bits of paper. You may
have seen balloons appearing to "stick" to walls, without any glue. You have probably seen
various types of plastic materials (such as thin plastic wrap) adhere to other objects, as if they
were "attracted" to each other across space. Magnets don't seem to be involved in this type of
phenomena. Could it be due to gravitational forces? Absolutely not, because gravitational forces
between these small objects is much too small to be noticeable.
In experiments using certain simple materials such as plastic and glass rods, when they
are rubbed with fur or silk, certain phenomena consistently recur: (1) when two of these objects
made from identical materials are prepared the same way (e.g., plastic rubbed with fur), the
objects appear to exert small repulsive forces on each other. If the objects are suspended by a
string, or mounted on a sensitive pivot, this force may result in a visible motion; (2) when two of
these objects made from different materials are held near each other (e.g., a plastic rod held near
a glass rod), the objects may also repel each other, but sometimes they appear to exert attractive
forces on each other. We will carry out these experiments in class. These forces are far too
strong to be related to the gravitational force, and, in any case, the gravitational force is never
repulsive, only attractive! So we must conclude that there is another force at work, which is
known as the "electrical" force. It may be attractive or it may be repulsive.
The experiments described above, and many other similar ones, lead us to conclude that
there are two different properties of matter that, in some sense, "cause" this force. When two
objects with property "A" are near each other, they push each other apart (i.e., exert repulsive
forces on each other). The same thing happens when two objects with property "B" are near each
other. But when an object with property "A" is near an object with property "B," they exert
attractive forces on each other. These properties of matter have been called "electric charge," and
instead of "A" charge and "B" charge, the terms "positive" [symbol:  ] and "negative"
[symbol:  ] charge are used. The symbols usually used for charge are ; or U, and it is
measured in units called "coulombs" [symbol: C]. The exact definition of the coulomb will be
deferred until later.
THE ELECTRIC FORCE
From the experiments with "charged" objects (i.e., objects with the charge property), we
are led to conclude that the magnitude of the electrical force depends strongly on the distance
between the objects. The effects of the repulsive and attractive forces are much more noticeable
when the charged objects are close together, than when they are far apart. Many careful
experiments have led to the following relationship for the magnitude of the electrical force
between two objects separated by a distance <, when one object has charge ;1 and the other
object has charge ;2 :
Ö
Jelectrical œ 5 |;1 | |;2 | Î <2
(Coulomb's law)
This relationship is called "Coulomb's law." Note how similar it looks, in mathematical form, to
Newton's law of gravitation. In Coulomb's law the letter 5 represents a proportionality constant,
which has a specific numerical value that depends on the system of units being used. The
absolute value signs on the charge symbols are there because the magnitude of the force does not
depend on whether the charges are positive or negative  it only depends on the amount of
charge present on the objects. Note that the electrical force does not depend on the mass of the
objects, or any other property  only on the electric charge.
How do the signs of the electric charges influence the electric force? They affect the
direction of the force, not the magnitude. The direction of the force depends on the relative
types of the charges. If ;1 and ;2 both have the same type of charge  positive or negative 
then the force between them is repulsive, and is directed along a straight line connecting the two
charges (see figures on the next page, first and second lines). If one charge is positive is the other
is negative, then the force is attractive, but still directed along the line connecting the charges
(see figure on the next page, third line).
Two positive charges repel:
Two negative charges repel:
A positive and a negative charge will attract:
oŠ † † † † † Šp
o‹ † † † † † ‹p
Šp † † o‹
In the SI system of units, distance is measured in meters (m), charge is measured in
coulombs (C), and force is measured in newtons (N). In these units, the constant 5 has the value
8.99 ‚ 109 N m2 /C2 , which we can approximate (with an error of barely 0.1%) by the value
9 ‚ 109 N m2 /C2 . As an old proverb says, "Better is the enemy of good enough," and using 9
instead of 8.99 is good enough. So, for instance, the magnitude of the electrical force between
an object with a charge of 3 .C and one with a charge of 6 .C, separated by a distance of 4 m,
is:
J œ
Ð9‚109 N m2 / C2 ÑÐ3‚106 CÑÐ6‚106 CÑ
Ð4 mÑ2
œ 1.0 ‚ 102 N
Electrical forces  like all other types of forces  obey the "superposition" principle.
This states that the net electrical force acting on any charged object is equal to the vector sum of
all the individual forces on that object, where each individual force results from the interaction of
the object and one other charged object. Each individual interaction is unaffected by any of the
other interactions. Algebraically, this can be expressed as:
Jt net œ Jt 1  Jt 2  . . .  Jt 8
Here, the net force on the object is found from the vector sum of the forces from 8 other charged
objects. This equation implies that the net B component of the force equals the sum of the
individual B components, and that the net C component equals the sum of the individual C
components.
Example 1. Suppose each grid square below is one meter long, and suppose that the three
dots represent identical electric charges. What is the net force acting on the middle charge?
•
U1
•
U2
•
U3
Answer: Net force equals vector sum of [force from Q3 ] plus [force from Q1 ]. These
forces are represented by left and right arrows, respectively, in the diagram below:
oqqqqqqq • qp
Since all three charges are identical, we can use the same symbol U to represent all of
them; that is: U1 œ U2 œ U3 œ U. The electric force exerted on any charge by any other is a
repulsive force, so the force on charge U2 due to charge U1 is towards the right (away from U1 )
and has magnitude 5U1 U2 /(12 m)2 = 5U2 /(144 m2 ), and the force on charge U# due to charge
U3 is towards the left and has magnitude is 5U3 U2 /(6 m)2 = 5U2 /(36 m2 ) = 45U# /(144 m2 ), so it
is four times larger than the magnitude of the rightward-pointing force. These forces are directed
along the line connecting the charges, and so we can see that each force in this case only has an
B component  the C components are zero. The rightward pointing force has a positive B
component, and the leftward pointing force has a negative B component. Then the net B
component is equal to 5U2 /(144 m2 ) + [  45U2 /(144 m2 )] œ [  35U2 /(144 m2 )]. This would
be represented by an arrow pointing to the left (negative B direction) with a length of three grid
squares (three times the length of the rightward pointing force). That arrow would then represent
the net electrical force acting on U2 , the middle charge.
Example 2. Suppose an object with zero net charge (i.e., exactly equal quantities of positive
and negative charges) is located in the neighborhood of another object with zero net
charge. What will be the net electrical force experienced by either object?
Answer: There will be virtually no net electrical force on either object, because all of the
repulsive and attractive forces will cancel each other out (i.e., their vector sum will be nearly
zero). Show this by considering two objects, each containing two protons and two electrons, and
drawing all of the force vectors on all of the charged particles. You should be able to see that the
net force on each object is nearly zero, as long as they are not located too close together.
QUANTIZATION OF ELECTRIC CHARGE
Electric charge has an important and unusual property, called the quantization of electric
charge: Every charge yet discovered in nature has an electric charge which is an integral
multiple of / œ 1.6 ‚ 1019 C, which is referred to as the "elementary charge." Electric charge
always appears in quantities that are some positive or negative integer multiple of /, e.g. 35/,
17984/,  13/, etc., but never 3.5/ or  1/ or 14,4448.7/. We say that electric charge is
"quantized" in units of /. There are subatomic particles called quarks which are the constituents
of neutrons and protons, which are quantized in units of /Î3. However, they don't appear by
themselves in nature, only in combinations which have charges that are integral multiples of /.
Atoms are composed of smaller, "sub-atomic" particles, such as the proton (:), neutron
(8), and electron (/). It turns out that while the neutron has no net charge (and so does not
experience electrical forces), both the proton and the electron have an amount of charge whose
magnitude is /. However, the proton has  / (a positive charge) and the electron has  / (a
negative charge). The fact that the charges on these two particles are the same magnitude is
interesting, in view of the very large mass difference between them (the proton has a mass nearly
2000 times larger than that of the electron). Physicists today have no verified explanation of this
large and curious mass difference between the basic negative and positive particles.
CONSERVATION OF ELECTRIC CHARGE
Electric charge exhibits another very important principle called "conservation of electric
charge." This principle states that the total amount of charge in any closed system never
changes. A "closed" system is one in which charges can neither enter nor leave. By "total
amount of charge," we mean the algebraic sum of all positive and negative charge quantities.
Example: If a sealed box initially contains positive charges equal to 6/, and negative
charges equal to  11/, the total amount of charge in that box must always remain at
6/  11/ œ  5/. It is possible that some of the positively charged particles and some of the
negatively charged particles may actually disappear  such phenomena do occur (and energy is
then given off in some form)  but nonetheless, the charge on the particles that do remain must
always sum up to  5/, and never anything else. So if later the positive charges equal to a total
of 96/, we know that the negative charges must equal a total of  101/.
QUESTIONS
Important Physics Concepts to use:
Positive and negative charges; Coulomb's law: J œ 5;1 ;2 Î<2
Protons (+) and electrons (-)
Superposition principle: Jt net œ Jt 1  Jt 2  ÞÞÞ  Jt 8
Vector addition: JnetßB œ J1B  J2B  ÞÞÞ  J8B
Newton's second law, +t œ Jt net Î7
On the graph below, an electric charge ;1 is at the origin and an electric charge ;2 is on the
positive B axis at B œ 5.0 m. Assume that there are no other charges anywhere nearby.
Use this graph for questions 1, 2 and 3.
qqqqqqoqqqqqqqoqqqqqqq
;1
;2
Question 1. If ;1 is positive and ;2 is negative, what is the direction of the electrical force
on ;1 ?
Ð1Ñ
(3)
(5)
(6)
in the positive B direction
(2) in the negative B direction
in the positive C direction
(4) in the negative C direction
the force is not directed precisely along any of the coordinate axes, but at some angle
there is no force in this case
(continued) . . .
Question 2. If ;1 and ;2 are both positive, what is the direction of the electrical force on
;1 ?
Ð1Ñ
(3)
(5)
(6)
in the positive B direction
(2) in the negative B direction
in the positive C direction
(4) in the negative C direction
the force is not directed precisely along any of the coordinate axes, but at some angle
there is no force in this case
Question 3. If ;1 and ;2 are both negative, what is the direction of the electrical force on
;1 ?
Ð1Ñ
(3)
(5)
(6)
in the positive B direction
(2) in the negative B direction
in the positive C direction
(4) in the negative C direction
the force is not directed precisely along any of the coordinate axes, but at some angle
there is no force in this case
Question 4
Question 5
Question 6
Question 4. A proton is located at the origin, and an electron is located at the point (3 m,
3 m). Sketch this on the diagram above and determine the direction of the electrical force
on the proton. (Suggestion: place the origin near the middle of the diagram, draw B and C
axes on your sketch, and let each grid mark represent 1 m.) What is the direction of the
electrical force on the proton?
(1) ß
(2) Å
(3) â
(4) Ã
(5) á
(6) à
What is the direction of the electrical force on the electron?
(1) ß
(2) Å
(3) â
(4) Ã
(5) á
(6) à
Question 5. Repeat question 4 using two protons at the origin and at the point (3 m, 3 m).
The vector representing the electrical force on the proton at the origin makes what angle
with respect to the positive B axis?
(1Ñ 0 °
(2) 45 °
(3) 90 °
(4) 135 °
(5) 225 °
(6) 270 °
Question 6. A proton is located at (0 m, 5 m) and an electron is located at the origin. The
electrical force on the electron is:
(1) directed toward the positive-C direction, and has greater magnitude than the electrical force
acting on the proton.
(2) directed toward the positive-C direction, and has smaller magnitude than the electrical force
acting on the proton.
(3) directed toward the positive-C direction, and has magnitude equal to the electrical force
acting on the proton.
(4) directed toward the negative-C direction, and has greater magnitude than the electrical force
acting on the proton.
(5) directed toward the negative-C direction, and has smaller magnitude than the electrical force
acting on the proton.
(6) directed toward the negative-C direction, and has magnitude equal to the electrical force
acting on the proton.
Question 7. Two charged particles are separated by a certain distance, and exert an
electrical force on each other. What will happen to the magnitude of this electrical force if
the separation between the particles is decreased?
(1) The force will decrease in magnitude.
(2) The force will increase in magnitude.
(3) The magnitude of the force will not change.
(4) The magnitude of the force may decrease or increase, depending on whether the
charges are like or unlike.
(5) The magnitude of the force on one particle will increase, while that on the other
particle will decrease.
Question 8. Two particles with charges ;1 and ;2 are separated by a distance <. There are
no other charges nearby. Consider the following actions:
I. increase ;1
II. increase ;2
III. increase <
IV. decrease <
Which of the above actions will cause the magnitude of the electric force between the
charges to increase?
(1) I and III only
(4) II and IV only
(2) I and IV only
(5) I and II and III
(3) II and III only
(6) I and II and IV
Question 9. When two charged particles are separated by 2 meters, the magnitude of the
electrical force between them is J . What will be the magnitude of this force if their
separation is increased to 4 meters?
(1) ¼ J
(2) ½ J
(3) J
(4) 2J
(5) 4J
(6) not enough information is given to determine magnitude of force.
Question 10. Which of these will result in the repulsive force between two identical
charged particles increasing by a factor of 8:
(1)
(3)
(4)
(5)
(6)
double one of the charges
(2) double both of the charges
double one of the charges and cut the particle separation in half
triple one of the charges and cut the particle separation in half
triple both of the charges
double both of the charges and double the particle separation
Question 11. A 6-C charge and a 12-C are separated by 2 m; there are no other charges
present. Compared to the electrical force on the 6-C charge, the electrical force on the 12-C
charge is:
(1) one-fourth as strong
(4) two times as strong
(2) one-half as strong
(5) four times as strong
(3) the same magnitude
Question 12. Suppose three electrons are equally spaced along the C axis as shown in the
diagram. The direction of the net electrical force on the middle electron is:
C
|
‹
|
‹
|
‹
|
(1) towards positive B
(2) towards positive C
(3) towards negative B
(4) towards negative C
(5) nowhere, since there is no net force on
the electron at the origin
Question 13. As shown in the diagram, protons are located on the B axis at B œ  2 m,
B œ 0, and B œ 1 m. The direction of the net electrical force on the proton at the origin is
(1) towards positive B
qq Š qqqq Š qq Š qq
2m
0
1m
(2) towards positive C
(3) towards negative B
(4) towards negative C
(5) nowhere, since there is no net force on
the proton at the origin
Question 14. In this figure, particles with charges  ;ß  ;ß  ; are located on the B
axis as shown. The direction of the net electrical force on the middle charge:
qq ‹ qq ‹ qqqq Š qq
(1) towards positive B
(2) towards positive C
(3) towards negative B
(4) towards negative C
(5) nowhere, since there is no net force on
the positive charge at the origin
Question 15. A electron is fixed at the origin; there are no other charges present. If a
negative charge is brought in and released at a nearby point, and allowed to move freely,
then as it moves the magnitude of the force acting on this negative charge will:
(1) always be zero
(2) remain constant, but nonzero
(3) always increase
(4) always decrease, but never reach zero
(5) sometimes increase and sometimes decrease (6) not enough information to decide
Question 16. A electron is fixed at the origin; there are no other charges present. If a
negative charge is brought in and released at a nearby point, and allowed to move freely,
then as it moves the magnitude of the acceleration of this negative charge will:
(1) always be zero
(2) remain constant, but nonzero
(3) always increase
(4) always decrease, but never reach zero
(5) sometimes increase and sometimes decrease (6) not enough information to decide
Question 17. A electron is fixed at the origin; there are no other charges present. If a
negative charge is brought in and released at a nearby point, and allowed to move freely,
then as it moves the speed of this negative charge will:
(1) always be zero
(2) remain constant, but nonzero
(3) always increase
(4) always decrease, but never reach zero
(5) sometimes increase and sometimes decrease (6) not enough information to decide
Question 18. A positive charge is located at the origin, and another positive charge is
located at the point (  4 m, 4 m). Sketch this situation on the diagram. The B component
of the electrical force on the charge at the origin is:
Ð1Ñ greater than zero
Ð2Ñ equal to zero
Ð3Ñ less than zero
Ð4Ñ may be equal to, less than, or greater
than zero, depending on the precise
magnitude of the two charges.
Question 19. Two protons are separated by two meters. Determine the magnitude of the
electrical force that they exert on each other, as well as the magnitude of their mutual
gravitational force. (Answer to one significant figure, with the correct power of 10.)
Electrical force =
Gravitational force =
What is the ratio of the magnitude of the electrical force acting between them compared to
the magnitude of the gravitational force acting between them? (Answer to one significant
figure, with the correct power of 10.)
Electrical force/Gravitational force =
Question 20. Three positive charges are sitting on the B axis, as shown in the figure below.
A 2-C charge is at B œ  2 m; a 5-C charge is at the origin, and a 6-C charge is at B œ 4
m. In black, draw and label two arrows representing the electrical forces on the charge at
the origin due to the other two charges. Label the force due to the 2-C charge "F2 ", and
the other one "F6 ". In red, draw an arrow representing the net electrical force on the
charge at the origin. Make sure the lengths of the arrows correspond to the relative
magnitudes of the forces.
qqqq ‹ qqqq ‹ qqqqqqqq ‹ qqqq
2C
5C
6C
Question 21. A 6-.C charge and a particle with a 9-.C charge are separated by 3 m.
Suppose they are now moved to a separation of 1 m; the repulsive force on the 6-.C charge
will now be different from the original value. The force on the 6-.C charge will go back to
the original value if the charge on the other particle is changed to what value?
New value of 9-.C charge = _____________________________________________
Question 22. A 1-C charge is located at the origin and another 1-C charge is located at the
point (1 m, 1 m). What are the B and C components of the electrical force (JB and JC ) on
the charge at the origin?
JB =
JC =
Question 23. A 2-µC charge and a 3-µC
charge are placed near to each other, and
then allowed to move apart (as they repel
each other). On the grid shown, draw an
accurate graph of the magnitude of the
electrical force that the charges exert on each
other.
Let the C-axis represent the magnitude of the force, and the B-axis represent
separation distance. Plot the graph for separation distances from 1 m to 10 m, at intervals
of 1 m. Plot (on the same graph!) the magnitude of the force acting on the 2-µC charge with
a red line, and that for the 3-µC charge with a blue line.
Make sure you label all your axes with appropriate units!
Topic 2B. Electric Fields
THE ELECTRIC FIELD AND WHAT PRODUCES IT
The influence of electrical forces is seen when various objects (such as plastic or rubber
rods, or tapes) are observed to attract and repel each other. We have described these objects as
being "charged." It has been found that the charge on these objects ultimately is due to very
small subatomic particles, called protons and electrons. Protons have positive charge, while
electrons have negative charge. (The other important subatomic particle, the neutron, is
uncharged, or "neutral," and does not experience electrical forces.) When we charge tapes or
other objects, the sources of the charge are the protons and electrons in the atoms and molecules
composing the objects.
Charged particles "alter" space. Charged particles exert strong forces on each other,
either attractive or repulsive; these are what we call "electrical" forces. The magnitude of the
force depends on the amount of the charge. Although the forces are exerted even when the
particles are far apart from each other, the effect does not happen instantaneously. Careful
measurements show that it takes a certain (small) amount of time for the charges to influence
each other. Apparently, each charged particle produces some change in the space surrounding it,
and this "alteration" in space eventually results in a force being experienced by any other
charged particle that is present in the neighborhood. The alteration "permeates" space, in that it
is apparently present everywhere. No matter where you place a charged particle in the space
surrounding another charged particle, it will experience a force as a result. But the force is
stronger when one is closer to the charged particle that is the "source" of the alteration in space.
This alteration is called the "electric field." This "alteration" in space that is produced
by a charged particle is not a material object (it doesn't have mass) and is ordinarily invisible.
However, it is a real physical entity; it has properties such as energy and momentum, and may be
detected in many different ways. This entity has been given the name "electric field" (symbol:
E), and it is a vector quantity: it has both magnitude and direction. We can detect its presence by
observing that a charged particle placed in an electric field will always experience a force which
will tend to "push" (or pull) that charged particle in some particular direction.
Electric fields are produced by particles with electric charge; a charged particle is always
involved at some stage in the production of an electric field. The charged particles responsible
for the production of the field are called "source" charges. When we place a charged particle in
an electric field to detect the presence of the field, or to measure the strength (magnitude) of the
field, the particle we use is called a "test" charge. We always assume that the test charge is so
small that it has no influence on the source charges, and that therefore the test charge does not
alter the existing electric field in any way. (If the test charges were too large they might alter the
location of the source charges, and in so doing they would change the electric field.) Of course,
the test charges produce their own electric field, but we assume that this field is much smaller
than the field produced by the source charges, and so we can ignore it.
CALCULATING AND USING THE ELECTRIC FIELD
The four cases described below are critical to understanding the nature of electric fields. They
show how to calculate or use the electric field in different situations. Make sure you understand
when to use each method described.
A. Determining the electric field produced by a single "point" charge: A point charge U
(i.e., an extremely small particle containing a charge of U) will produce an electric field at every
point P in space;
a. the magnitude of the field it produces at point P is given by this relationship:
I œ 5lUlÎ<# Þ where 5 œ 9 ‚ 109 N † m2 /C2 and < is the distance from the charge U to the point
P.
b. The direction of the field it produces at P is directly away from U if U is a positive
charge, and it is directly toward U if U is a negative charge. In either case, it is directed along
the straight line connecting U to the point P.
t
Š - - - - - - - - - - <- - - - - - - - - - - - - • qqqp I
P
U
B. Determining the total electric field at a point due to many point charges: The net electric
field at a point in space is equal to the vector sum of all of the individual electric fields at that
point, produced by all source charges present. Mathematically, this is expressed by this equation:
It œ It 1  It 2  ÞÞÞ  It 8
Here, It 1 is the electric field at point P produced by charge U1 ; It 2 is the electric field at
point P produced by charge U2 , and so on. (We are assuming that there are 8 source charges,
labeled U1 on up to U8 .) Each of these individual electric fields is determined as in case (A).
C. Determining the magnitude and direction of the electric field experimentally. The
magnitude and direction of the electric field at a particular point in space can be determined
experimentally by placing a "test" charge at this point. A "test" charge is a charge so small [small
amount of charge, not necessarily small in size] that it has no influence on the electric field itself.
The amount of charge on the test charge is symbolized by ;test .
a. The magnitude of the electric field at a point P is given by this equation:
I œ J;test Î;test
where J;test is the magnitude of the force experienced by the test charge (;test ) at the point P.
b. The direction of the electric field at a point P is defined to be the same as the
direction of the force experienced by a positive test charge placed at that point. (If a negative
test charge is used instead, the electric field is in the direction opposite to that of the electric
force on that negative charge.)
D. Determining the electric force that acts on a charged particle, knowing the electric field
at its position. Suppose that we know the electric field at some point in space from A, B, or C
above  it doesn't matter how we know it, but evidently there are some "source" charges whether
we know them or not. If any particle with charge ; is located at that point in space then the
particle experiences a force.
a. The magnitude of the force is given by this equation: J œ l;l I . It is the product of
the magnitude I of the electric field at that point in space and the absolute value l;l of the
charge. (So a charge  ; and a charge  ; will experience a force of the same magnitude, at a
given point in space.)
b. The direction of the force is:
i. in the same direction as the electric field direction, in the case of a positive charge;
ii. in the direction opposite that that of the electric field, in the case of a negative
charge.
Review Questions:
1. If you have a test charge ; and know nothing about what other charges exist or what the
electric field is anywhere, what can you do if you are able to measure the electric force on the
test charge anywhere?
Answer: Place the test charge at different locations, determine the electric force acting on it at
each location, and from that determine the electric field (magnitude and direction) at each point.
2. If you know nothing about what charges exist, but know the value of the electric field at some
location, what can you do?
Answer: You can predict the electric force that will act on any test charge you place at that
location. Just use Jt œ ;It .
Source charges "present" or "absent" (out of the picture)?
A very confusing aspect of problems involving the electric field is that sometimes we
have to explicitly consider the detailed properties of the source charges  the magnitude of
their charge, their location, etc.  and at other times we completely ignore the source charges.
You need to learn to recognize when you need to consider the source charges and when you
don't.
Source charges absent: Sometimes the source charges are "out of the picture," and we
have no information about them except that there is an electric field present, and we know that
the source charges that produced it must be somewhere. In this type of problem, we are given
information about the electric field that is present, but not about the source charges which
produced it. Instead, our attention will be focused on the test charges, and on how they are
affected by the electric field. (When the source charges are "out of the picture," we sometimes
say that there is an "external" electric field present. This means that the electric field is produced
by charges that are "external" to the region that is being shown.)
Source charges present: In other types of problems involving the electric field, we are
given information about the source charges and we have to take this information into
consideration. For instance, we may have to determine the magnitude and direction of the
electric field at a certain point in the neighborhood of one or more source charges. In order to
solve a problem of this type, we need to remember the formula for the magnitude of the electric
field at a point in space near a single charged particle. The magnitude E of the electric field at a
point a distance < from a particle with charge U is given by I œ 5lUlÎ<2 . (5 œ 9 ‚ 109 N †
m2 /C2 .)
How do you know whether source charges are present or absent? The wording of
each problem should allow you to decide if it is a "source charges present" problem, or if instead
the source charges are absent, and "out of the picture."
1) If the problem gives explicit information about the electric field  for instance, that
it is uniform, or "strong," or points in some particular direction  it is very likely that you do
not have to worry about the source charges. You only have to figure out what would happen to
different electric charges when placed somewhere in the field, using Jt œ ;It .. (E.g.: What
forces are acting on the charges? Where will they move?)
2) If the problem gives information about the magnitude and location of one or more
charged particles, and asks you to figure out something about the electric field at a particular
location, then "source charges are present." You will probably have to find the electric field
vector due to each source charge, and find the vector sum to get the "net" field at a particular
point, and then you can use this information to determine the force on a test charge placed at that
point. The magnitude of the field at a point due to a source charge U is given by I œ 5lUlÎ<2
where < is the distance from the point to the source charge. The direction of the field is along the
line connecting the point and the charge, either away from the source charge if it is positive, or
toward it if it is negative. Note that if the source charges are not present in the problem you can
not make use of the equation I œ 5lUlÎ<2 , because you have no way of knowing the value of <!
Example 1: A proton is located at the point
(  1 m, 0 m) and two protons are located at
(  1 m, 0 m). What is the magnitude of the
net electric field at the origin?
(A) 0 N/C
(D) 1.08 ‚ 109 N/C
(G) 4.32 ‚ 109 N/C
(B) 0.36 ‚ 109 N/C
(E) 1.44 ‚ 109 N/C
C
qqqp• qqq qqqp• qqqq B
(C) 0.72 ‚ 109 N/C
(F) 2.88 ‚ 109 N/C
Solution: In this problem you are asked to find the net electric field at the origin, with
three charged particles present (all protons). There will be three separate field contributions, one
for each "source" charge; you have to find the vector sum of all three. The magnitude of each of
the three fields is the same; it is:
I œ (9 ‚ 109 N † m2 /C2 ) (1.60 ‚ 1019 C) / (1 m)2 = 1.44 ‚ 109 N/C.
One field is directed toward the positive B axis, two are directed toward the negative B
axis. We could write the result this way À
InetßB œ I1B  I2B  I3B
= (1.44 ‚ 109 N/C)  (  1.44 ‚ 109 N/C)  (  1.44 ‚ 109 N/C)
œ 1.44 ‚ 109 N/C.
The C component of the net field is zero, and so the magnitude of the net electric field is equal to
the magnitude of the B component of the net field, which is 1.44 ‚ 109 N/C. (So the correct
answer is choice E.)
Example 2: A uniform electric field is present in a certain region of space. In order to
measure the strength of this field, you place a particle with charge ; at some point and
measure the force on that particle. If you remove the charge ; , and instead you place a
charge 2; at that same point, the electric field magnitude (not the force) you now measure
at that point will be:
(A) the same.
(D) half as large.
(B) twice as large.
(E) one quarter as large.
(C) four times as large.
Solution. In this problem, you are told that "a uniform electric field is present in a
certain region of space." From this you should conclude that the field will always be uniform,
and will not be influenced by any charges you place in the field. You are not given any
information about the source charges, and so you must assume that the field is a fixed "external"
field. The source charges will not be influenced by any other charges that might be brought in,
and the field has a given magnitude and direction at every point in space which will not be
affected in any way by a test charge. If you now bring in a test charge ; to determine the
magnitude and direction of the field at some point, the test charge will experience a force whose
magnitude is given by J; œ ;I . The magnitude and direction of I are determined by the source
charges; they have a specific value at each point in space, and they will not be changing. If,
however, you put a test charge 2; at the point where you had originally placed the ; charge, the
force on this new test charge will have twice the magnitude as that on the original test charge:
J2; œ 2;I . The field at that point did not change, just the magnitude of the force on the
(different) test charge. So, the correct answer is A, "the same."
Example 3: A particle with charge  ; and mass 7 is held at a certain point in apparently
empty space, and then released to move freely. It is observed to move with initial
acceleration a toward the north. If a particle with charge  ; and mass 27 is now placed
at the same point, it will:
ÐAÑ move north with acceleration +
(C) move north with acceleration 2+
(E) move south with acceleration 0.5+
(B) move north with acceleration 0.5+
(D) move south with acceleration +
(F) move south with acceleration 2+
Solution. In this problem, you are told that a charged particle is held at a certain point in
"apparently empty space," and when it is released it begins to accelerate. This tells you that there
must be an electric field present at the point where it is released. Once again you are not given
any information about the source charges, and so you can only determine the properties of the
electric field by "probing" it with various test charges. As discussed above, if a positive charge
located at some point in space experiences a force in some direction, then a negative charge at
that same point will experience a force in the opposite direction. (Just consider, for instance, a
point in the neighborhood of a positive source charge fixed at the origin. Put a positive test
charge at that point, and it will move away from the origin; put a negative test charge there, and
it will move toward the origin.)
The magnitude of the force on a test charge in an electric field is given by J œ |; | I . The
magnitude of the force on a charge  ; is the same as that on a charge  ; ; only the directions
are different (opposite). So here, the magnitude of the force on the two charged particles is the
same; if the positive charge is forced to move north, then the negative charge will accelerate
toward the south. However, the magnitude of the acceleration is proportional to the ratio of
[force] Î [mass]. This means that the acceleration of the more massive particle (mass œ 27)
will be half that of the less massive particle (since the force is the same for both). Therefore, the
particle with charge  ; has acceleration +Î2, and moves toward the south. The correct answer
is E.)
Example 4: At a particular point in otherwise empty space, a 4.6-C charge is placed and it
experiences an electrical force of 14.8 N. If this charge is removed, and instead a 6-C
charge is placed at that point, what will be the magnitude of the force that it will
experience?
Solution. Once again, a charged particle is placed at a point in "otherwise empty space,"
and it experiences a force. There must be an electric field present, and we can find the magnitude
of this field. It is given by I œ J Î; œ 14.8 NÎ4.6 C œ 3.22 N/C. If we now place a 6-C charge
at the same point in space, it will be influenced by the same electric field, and so the force on this
particle will be given by J œ ;I œ (6 C) (3.22 N/C) = 19.3 N.
Note that nowhere in problems #2, #3, or #4 could you use the equation I œ 5UÎ<2 ,
because at no time did you have any information about any source charge U. (In fact, there may
have been many source charges present, but you had no information about any of them.) The
source charges were present in #1, but you could also use your result to determine the force that
would act on a test charge (e.g., ; œ  /) placed at the origin in that situation.
ELECTRIC FIELD LINES
The electric field in a region can be nicely visualized using "electric field lines" which
indicate the direction and strength of the electric field at any point in space. While the electric
field is a real physical quantity, the electric field lines are just abstractions.
Electric field lines are continuous non-intersecting lines which originate on positive
charges and terminate on negative charges, or else go to or come from infinity. The direction of
the electric field at a point in space is tangent to the electric field line that passes through the
point, and the magnitude of the electric field is proportional to the density of lines around that
point.
The next page gives the rules for drawing electric field lines and shows some examples
for different situations.
The rules for drawing electric field lines are the following:
(1) Electric field lines originate on positive charges and terminate on negative charges. They do not
begin or end in regions without electric charge, so the electric field lines are continuous in charge-free
regions of space. However, electric field lines may continue to infinity or come from infinity, if the
system of charges is not neutral.
(2) The number of electric field lines originating or terminating at the electric charges is proportional to
the magnitude of the electric charge. For example, if a system consists of charges U, 5U, and  6U,
one might draw 3 lines originating on U, 15 lines originating on 5U, and 18 lines terminating on  6U.
(3) Very close to an electric charge the field lines are radial (inward or outward) and are isotropically
distributed, their total number being proportional to the magnitude of the charge.
(4) The electric field vector at a point is tangential to the electric field line at that point.
(5) Electric field lines do not touch or cross.
(6) The distance between the electric field lines in a region is proportional to the magnitude of the
electric field in that region.
The diagram on the left shows the electric field lines in the vicinity of a positive charge,
and the diagram on the right shows those near a negative charge. In both cases the lines are
directed straight away from or towards the charge, and are evenly spaced. The negative
charge has twice the magnitude of the positive charge, so it has twice as many lines.
Left: The electric field lines near a system of two identical positive charges. Middle: The
electric field lines near a system of a positive and a negative charge of the same magnitude.
Right: The electric field lines near a system of a positive charge  2; and a negative
charge  ; with half the magnitude; in this case half the field lines originating on the positive
charge terminate on the negative charge, and the other half go out to inifinity as they would
for a single charge  ; .
In a region with a constant electric field, the electric field lines are particularly simple:
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
qqpqqpqqpqqpqqpqqpqqpqqp
Now, suppose that an irregular piece of conducting material (say a metal ingot) is placed
in this constant electric field. What happens?
First of all, the charges in the conductor rearrange themselves. The negative charges will
experience a force to the left (opposite to the direction of the electric field) and the positive
charges will experience a force to the right (in the direction of the electric field).
The electric field in the region of the conductor is then changed because there is a change
in the charge distribution. Some of the field lines now terminate on the negative charges on the
left side of the conductor, and an equal number of field lines now originate on the positive
charges on the right side of the conductor. An example of what this might look like is shown
below. Note that the electric field lines are significantly changed in the vicinity of the
conductor, but not far away from it.
QUESTIONS
Question 1. Which of these is a sure indication that there is an electrical field present
everywhere in a certain region of otherwise empty space?
Ð"Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
(5)
A charged particle anywhere in the region always remains motionless.
A charged particle anywhere in the region always moves with constant velocity.
A charged particle in the region never changes its direction of motion.
A charged particle anywhere in the region always experiences a nonzero electrical force.
None of the above is a sure indication of the presence of an electric field.
Question 2. How can you determine whether an electrical field is present in a particular
region of space which appears to be completely empty?
Ð1Ñ Shoot an uncharged particle into the region and see whether it speeds up, slows
down, or changes direction.
Ð2Ñ Shoot a charged particle into the region and see whether it speeds up, slows down, or
changes direction.
Ð3Ñ Hold a charged particle just outside the region and see whether it experiences an
attractive or repulsive force.
Ð4Ñ Move a charged particle into the region and see whether its charge increases or
decreases.
Ð5Ñ There is no way to determine whether an electric field is present in a region of space
that appears to be completely empty.
Question 3. A proton is placed at the origin, and then released and allowed to move freely.
It begins to move along the positive B axis. From this one can conclude that the electric
field at the origin points:
Ð1Ñ
Ð3Ñ
Ð5Ñ
Ð6Ñ
toward the positive B direction.
(2) toward the negative B direction.
toward the positive C direction.
(4) toward the negative C direction.
in a direction not along either the B or C axis.
There is not enough information to determine the electric field direction at the origin.
Question 4. An electron is placed at the origin, and the released and allowed to move freely.
It begins to move along the positive B axis. From this one can conclude that the electric
field at the origin points:
Ð1Ñ
Ð3Ñ
Ð5Ñ
Ð6Ñ
toward the positive B direction.
(2) toward the negative B direction.
toward the positive C direction.
(4) toward the negative C direction.
in a direction not along either B or C axis.
There is not enough information to determine the electric field direction at the origin.
Question 5. In a certain region of space, the electric field is zero everywhere. This means
that, if a charged particle is located anywhere in this region:
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
the particle can not be moving.
the particle experiences no net electrical force.
the particle experiences a repulsive electrical force.
the particle experiences an attractive electrical force.
the particle is always forced back toward one particular location.
Question 6. Throughout a certain region of space, the electric field has uniform magnitude
and direction. This means that wherever a particular charged particle is placed at rest in
this region and then allowed to move freely, it will always:
i. remain motionless.
ii. move with constant speed.
iii. move in an unchanging direction.
iv. move with constant magnitude of acceleration.
Ð1Ñ i only
Ð2Ñ ii only
Ð3Ñ ii and iii only
Ð4Ñ iii and iv only
Question 7. In the neighborhood of an isolated charged particle, the electric field:
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Ð6Ñ
has uniform magnitude and direction.
has uniform magnitude, but points in many different directions.
has uniform direction, but greater magnitude near the particle.
has uniform direction, but smaller magnitude near the particle.
points in many different directions, and has greater magnitude near the particle.
points in many different directions, and has smaller magnitude near the particle.
Question 8. When a charge ; is placed at a certain point in an electrical field, it
experiences a force toward the west of magnitude J . If instead a charge 2; were placed at
this same point, what force would it experience?
Ð1Ñ an eastward force of magnitude J Î2
Ð3Ñ an eastward force of magnitude 2J
Ð5Ñ a westward force of magnitude J
Ð2Ñ an eastward force of magnitude J
Ð4Ñ a westward force of magnitude J Î2
Ð6Ñ a westward force of magnitude 2J
Question 9. When a charge ; is placed at a certain point in an electric field, it experiences a
force toward the west of magnitude J . If instead a charge  ;Î2 were placed at that same
point, what force would it experience?
Ð1Ñ an eastward force of magnitude J Î2
Ð3Ñ an eastward force of magnitude 2J
Ð5Ñ a westward force of magnitude J
Ð2Ñ an eastward force of magnitude J
Ð4Ñ a westward force of magnitude J Î2
Ð6Ñ a westward force of magnitude 2J
Question 10. An electron is placed at the origin. The direction of the electric field at the
point B œ 0 m, C œ 3 m, is:
Ð1Ñ
Ð3Ñ
Ð5Ñ
Ð6Ñ
toward positive B
Ð2Ñ toward positive C
toward negative B
Ð4Ñ towards negative CÞ
not along either the B or the C axes.
There is not enough information to determine the direction of the electric field.
Question 11. Electrons are located on the C axis at C œ 3 m and C œ  3 m. The direction
of the net electric field at the origin is:
Ð1Ñ
Ð3Ñ
Ð5Ñ
Ð6Ñ
toward positive B
Ð2Ñ toward positive C
toward negative B
Ð4Ñ towards negative C
not along either the B or the C axes.
There is not enough information to determine the direction of the electric field.
Question 12. Electrons are located at the points ÐB œ  1 m, C œ 0 mÑ and ÐB œ 1 m,
C œ ! m). The direction of the net electric field at the point ÐB œ 0 m, C œ 1 m) is:
Ð1Ñ
Ð3Ñ
Ð5Ñ
Ð6Ñ
toward positive B
Ð2Ñ toward positive C
toward negative B
Ð4Ñ towards negative C
not along either the B or the C axes.
There is not enough information to determine the direction of the electric field.
Question 13. A proton is placed at the origin. The direction of the electric field at the point
B œ 2 m, C œ 2 m, with respect to the positive B axis, is:
Ð1Ñ 30°
Ð2Ñ 45°
Ð3Ñ 120°
Ð4Ñ 135°
Ð5Ñ 210°
ÐFÑ 225°
Question 14. A strong electric field in a certain region points toward positive B. An
electron is released at the origin, and then a long time later a proton is released at the
origin. When released,
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Ð6Ñ
the electron will move toward positive B, and the proton will move toward negative B
the electron will move toward negative B, and the proton will move toward positive B
the electron will move toward positive C, and the proton will move toward negative C
the electron will move toward negative C, and the proton will move toward positive C
both the electron and the proton will move toward positive B
both the electron and the proton will move toward negative B
Question 15. An electron is located at the origin as shown. Which arrow best describes the
direction of the electric field at point A?
(1) Å
(2) Æ
(3) ß
(4) â
(5) à
(6) á
Question 16. A proton is located at the origin as shown. The electric field is shown at a
nearby point. Which arrow best fits the electric field vector at point B?
(1)
(3)
â
â
(5) â
(2) à
(4)
à
(6) à
Question 17. Which of these statements about electric fields is not true?
(1) The electric field at a point can be completely described by a number, which corresponds to
the magnitude of the electric field.
(2) The electric field at a point would exert the same magnitude of force on an electron placed at
that point, as it would on a proton placed at the same point.
(3) Electric fields do not exert forces on uncharged particles.
(4) Electric fields can exist in a vacuum (i.e., in "empty space").
(5) Electric fields are produced by electric charges.
Question 18. Charge A is 2 C and charge B is  4 C. They are sitting in a uniform electric
field. Which of these diagrams correctly shows the forces that are exerted on charges A and
B by the electric field ( not the forces they exert on each other)?
(1)
A Š qqp
oqq ‹ B
(2)
A Š qqp
oqqqqq ‹ B
(3)
oqq Š A
B ‹ qqp
(4)
oqqqqq Š A
B ‹ qqp
(5)
A Š qqp
B ‹ qqp
(6)
A Š qqp
B ‹ qqqqqp
Question 19. Which of these charges is experiencing the electric field with the largest
magnitude?
Ð1Ñ
Ð2Ñ
(3)
(4)
(5)
(6)
A 2-C charge acted on by a 4-N electric force
A 3-C charge acted on by a 5-N electric force.
A 4-C charge acted on by a 6-N electric force.
A 2-C charge acted on by a 6-N electric force.
A 3-C charge acted on by a 3-N electric force.
All of the above are experiencing electric fields with the same magnitude.
The diagram to the right shows the electric
field lines in the vicinity of three charges, ;1 ,
;2 , and ;3 .
Question 20. The signs of these three
charges are, respectively,
(1) positive, positive, positive
(2) positive, positive, negative
(3) positive, negative, positive
(4) negative, positive, positive
(5) positive, negative, negative
(6) negative, positive, negative
Question 21. The relative magnitudes of these three charges are, respectively,
(1) 1 : 1 : 1
(2) 1 : 1 : 2
(3) 1 : 2 : 1
(5) 1 : 2 : 2
(6) 2 : 1 : 2
(7) 2 : 2 : 1
Ð1Ñ A 2-C charge acted on by a 4-N electric force
(4) 2 : 1 : 1
Topic 2C. Electric Potential Energy
The electrical force is a conservative force, so there is a potential energy function
associated with it.
We can now summarize the two general principles that will guide us in understanding
electric potential energy:
1) If a charged particle is being acted on only by electrical forces (no push or pull
by direct contact), then the total energy of the charged particle must remain constant. This
means that ?X I œ 0, and so ?OI  ?T I œ 0. This means that ?T I œ  ?OI , so an
increase in T I equals a decrease in OI (and vice-versa), and a decrease in T I equals an
increase in OI (and vice-versa).
2) If a charged particle is being acted on by both electrical forces and direct-contact
[nonconservative] forces, then the change in the total energy of the charged particle is
equal to the net work done by the nonconservative forces: ?X I œ [ non-cons.
We will now consider two basic physical situations: (1) a region where the electric field
is uniform (same magnitude and direction everywhere); and (2) a region in the neighborhood of a
single, isolated "source" charge. Most real situations are more complicated than these, but we
can understand most of the basic ideas by looking at these two situations.
I. Region where electric field is uniform (i.e., electric field has same magnitude and
direction everywhere in region).
Let's consider what happens to a positive test charge ; when we release it from rest in
this region. Its initial kinetic energy is zero, but as time goes on, the particle speeds up (due to
the electrical force acting on it) and so it gains kinetic energy. Since only the electrical force is
acting on it (we assume), its total energy must be constant. Therefore, its potential energy must
be decreasing. How much will its potential energy change if it travels a distance . ? The decrease
in potential energy must be equal to the increase in kinetic energy, and  according to the
Work-Energy theorem  this is equal to the work done on the particle by the electric force. If
the particle is released from rest, it will travel in the direction of the electric field (since that's the
direction of the force acting on it). The magnitude of the force acting on it is equal to ;I ¸ so we
have the following relationship:
 ?T I œ ?OI œ [ œ J . cos ) œ J . cos 0° œ J . œ ;I.
What this equation says is this: If a positive charge ; travels a distance . in the direction of a
uniform electric field with magnitude I , then the potential energy of this charge will decrease by
an amount equal to ;I. . (We can see that its potential energy decreases, because if ;I. is
positive, then ?T I will be negative: ?T I œ  ;I. ). That means that the final value of the
potential energy is less than the initial value, by an amount equal to ;I. . (If the charge does not
travel in a straight line, then the change in potential energy is given by ;I. cos ) where . is the
magnitude of the charge's displacement.)
Now, where is the potential energy, T I , equal to zero? The answer is that, as in
mechanics, we can set it equal to zero anywhere we want. This is because the quantity that really
has meaning is ?T I  the change in T I  not the absolute value of T I itself. However, we
usually choose some convenient location at which to set T I œ 0. Then, we will often speak of
the "T I " value of the charge at some point, even though what we really mean is the difference
in T I values between that point and the point where we have chosen to set T I œ 0.
Question #2: Suppose a charged particle, which was released from rest, loses 3 J of potential
energy while traveling a distance of 4 m in a uniform electric field. (a) How much potential
energy will it lose if it travels a distance of 12 m in this same field? (b) How much kinetic energy
will it gain if it travels a distance of 10 m in this same field?
Answers: 9 J; 7.5 J
II. Region in neighborhood of a single, isolated "source" charge U.
The situation in this case is more complicated, because the electric field is not uniform in
the neighborhood of an isolated source charge. The electric field at any point in that
neighborhood points away from a positive source charge (and toward a negative source charge).
Thus, the direction that the field points depends on where you are located. Also, the field has
larger magnitude near the source charge: remember that the magnitude is given by I œ 5 |U|/<2 .
Now, suppose we have a positive source charge U fixed in position at the origin, and we
place a positive test charge ; somewhere in the neighborhood, initially at rest, at a distance <0
from the source charge U. What will happen to the test charge when it is allowed to move
freely? It will begin to move away from the source charge (since it's repelled), and as it moves, it
will speed up. This is because since it always experiences a force, the force will cause it to
accelerate, and so its speed continuously increases. Since its kinetic energy must be increasing,
once again we see that its potential energy must be decreasing (because, again, its total energy
must remain constant). This tells us that the farther away it gets from the source charge, the
smaller will be its potential energy.
How much will the potential energy change? Again, because X I œ constant, we will
have:
?T I œ  ?OI œ  [ .
But in this case, we cannot use [ œ J . cos ) because the force is not constant. Using the
methods of calculus it is possible to find the work done on the charge by the electric force when
the charge changes its position. This leads to the following result:
5;U
5;U
?T I œ  ?OI œ  [ œ <  <
3
0
From this we see, as we expect, that as the charge ; moves away from U (meaning that
<0  <3 ) its potential energy decreases À since <0  <3 , ?T I will be negative, which means that
T I is decreasing.) Does its T I ever get down to zero? In a case like this, we usually decide to
set the value of T I equal to zero when the test charge is infinitely far away from the source
charge; this is when the value of < is infinity. (This choice usually makes problems of this type
easier to solve.) Given this choice, the potential energy T I of a test charge ; a distance < from a
source charge U will be given by the following simple equation:
Ê
TI œ
5;U
<
Note that for a given source charge U and a given test charge ; , the potential energy of
the charge ; depends only on its position. It does not matter how charge ; arrived at that position
or what path it followed.
Now, what if the test charge is negative, i.e., ;  0 (while the source charge U remains
positive)? That equation seems to say that the potential energy of a negative test charge would be
negative; does that make sense? Well, suppose we place a negative test charge at rest in the
neighborhood of U; what will happen? It will start to move in toward U (since it's attracted); so
its kinetic energy increases as < gets smaller. Or, to put it another way, its PE decreases as <
decreases and increases as < increases. But we have set T I œ 0 when < œ ∞; how could T I be
increasing up to a value of zero? Answer: By starting out at some negative value. Therefore, the
equation is actually correct for ;  0.
Now, what about for a negative source charge (U  0)? You should be able to devise an
argument to show that the same equation above works in that case as well.
Example 1: Two parallel metal plates are connected to a battery, which creates a uniform
electric field between the plates. The electric field points from the left-side plate to the
right-side plate. A 2-C charge is held at rest somewhere between the plates, and then
released. When it strikes the right-side plate, its kinetic energy is 6 J, and its potential
energy is 0 J. What was its potential energy just before it was released?
Solution: When it strikes the right side plate, the total energy (TE) of the charge is 6 J (6
J + 0 J). Since its total energy does not change, that must have been the original value of its total
energy as well. Initially, it was at rest so its kinetic energy was zero. Therefore, the initial value
of its potential energy must have been 6 J.
Example 2: A uniform electric field of magnitude 4 kN/C is set up in a region 8 m wide. A
2-.C charge is held at rest on a wall on one side of the region, and then released. What is
its kinetic energy when it strikes the wall on the other side?
Solution: The change in the kinetic energy is equal to the work done on the charge. The
charge is released from rest, and so will travel in the same direction as the electric field.
Therefore, we get
[ œ J . cos ) œ J . cos 0° œ J . œ ;I. œ Ð2 .CÑÐ4 kN/CÑÐ8 mÑ œ 0.064 J.
This tells us that the change in the kinetic energy was 0.064 J. Since the initial value of the KE
was 0 J, we see that the final value of the KE, when the charge strikes the wall, is 0.064 J.
Example 3: A 3-C charge is fixed at the origin. A 2-C charge is held 3 m from the origin,
and then released. Later, a 1-C charge is held 2 m from the origin, and it is released. All
three charged particles have the same mass. Which of these statements is true, in
comparing the 2-C and the 1-C charges:
(A) The 2-C charge will eventually attain the fastest speed (i.e., when it is very far from the origin.)
(B) The 1-C charge will eventually attain the fastest speed (i.e., when it is very far from the origin.)
(C) Both the 1-C and the 2-C charges will eventually attain the same speed.
Solution: To figure out which charge eventually attains the fastest speed, we have to
remember that the total energy of each charge will not vary as it moves, since only electrical
forces are acting. For each charge, the initial kinetic energy is zero, since they are released from
rest. Therefore, all of their energy is initially potential energy. Eventually, they will move very
far away from the origin due to the repulsive force. In fact, they will eventually move so far that
their potential energy is very close to zero. (Remember that their potential energy is given by
T I œ 5U;Î<. When < is very large, T I is very small.)
So, when they get very far from the origin, their potential energy is nearly zero and
nearly all of their total energy will be kinetic energy. Eventually, their kinetic energy will be
nearly equal to the amount of potential energy that they had at the moment they were released.
Since they both have the same mass, whichever has the largest final kinetic energy will also have
the fastest final speed.
So, all we have to do is figure out which one has the largest initial potential energy. The
initial potential energy of the 2-C charge is given by by T I œ 5Ð3 CÑÐ2 CÑÎÐ3 mÑ œ 25 C2 /m.
The initial potential energy of the 1-C charge is given by T I œ 5Ð3 CÑÐ1 CÑÎÐ2 mÑ œ 1.55
C2 /m. Therefore, the 2-C charge has the larger initial potential energy. So, it will eventually have
the largest kinetic energy, and therefore the fastest speed. This means the correct answer is A.
Example 4: An object with a mass of 2 kg and a net charge of 4 .C is shot from a gun
aimed at the origin. The gun is located 20 km from the origin; the initial speed of the object
is 3 m/s. A particle with a charge of 0.001 C is fixed at the origin. How close will the object
get to the origin before it slows to a stop and starts back the other way?
Solution: The initial total energy equals the final total energy:
OI (initial)  T I (initial) œ OI (final)  T I (final).
We know that OI (initial) œ "# 7@2 = (0.5)(2 kg)(3 m/s)2 = 9 J, and OI (final) = 0 J.
Also, T I (initial) is nearly zero, since T I œ 5U;Î< and <(initial) = 2 ‚ 104 m.
Then we have X I (initial) = X I (final) Ê 9 J + 0 J = 0 J + T I (final), so
T I (final) = 9 J. Then 5U;Î< œ 9 J, or
< œ 5U;ÎÐ9 JÑ = (9 ‚ 109 N m2 C2 )(0.001 C)(4 ‚ 106 C)ÎÐ9 JÑ œ 4 m.
Questions on Electric Potential Energy
Prerequisite Concepts
·
Positive and negative charge; Coloumb's law
·
Definition of electric field
·
Electric field of a parallel plate capacitor
·
Kinetic energy and mechanical potential energy
·
Definition of work; work/energy relationship
·
Conservative forces/conservation of total energy
·
Electrical force is conservative
[Note: All gravitational forces may be ignored in this chapter]
Š    •P      B
Questions #1-6 refer to this figure, which shows a positive charge that is fixed in position at the
origin. Suppose a positive charge ; is placed at position P, and then released so that it (the
charge ; ) is free to move.
1. What will happen to this charge ; ?
(1)
(2)
(3)
(4)
(5)
It will not move.
It will move closer to the origin.
It will move farther away from the origin.
It will start moving closer to the origin, but then reverse direction and start moving back out again.
It will start moving away from the origin, but then reverse direction and start moving back in again.
2. As the charge ; moves, what will happen to the magnitude of the electrical force acting
on it?
(1)
(2)
(3)
(4)
(5)
The force will remain constant.
The force will increase in magnitude.
The force will decrease in magnitude, but never quite reach zero.
The force will decrease in magnitude, and at a certain point will reach zero.
The force will begin to decrease in magnitude, but then will start to increase again.
3. As the charge ; moves, what will happen to its acceleration?
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
The acceleration will remain constant.
The acceleration will increase in magnitude.
The acceleration will decrease in magnitude, but never quite reach zero.
The acceleration will decrease in magnitude, but at a certain point will reach zero.
The acceleration will begin to decrease in magnitude, but then will start to increase again.
4. As the charge q moves, what will happen to its speed?
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Its speed will remain constant.
Its speed will always increase.
Its speed will increase for a while, and then remain constant.
Its speed will increase for a while and then will start to decrease, but never quite reach zero.
Its speed will increase for a while, and then will start to decrease until it comes to rest.
5. As the charge q moves, what will happen to its kinetic energy?
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Ð6Ñ
Its kinetic energy will remain constant.
Its kinetic energy will always increase.
Its kinetic energy will increase for a while, and then remain constant.
Its kinetic energy will increase for a while and then start to decrease, but will never quite reach zero.
Its kinetic energy will continuously decrease, until at some definite point it reaches zero.
Its kinetic energy will continuously decrease, but will never quite reach zero.
6. As the charge ; moves, what will happen to its electric potential energy?
Ð1Ñ Its electric potential energy will remain constant.
Ð2Ñ Its electric potential energy will always increase.
Ð3Ñ Its electric potential energy will increase for a while, and then remain constant.
Ð4Ñ Its electric potential energy will increase for a while and then will start to decrease, but will never
quite reach a constant minimum value.
Ð5Ñ Its electric potential energy will continuously decrease, until at some definite point it reaches a
constant minimum value.
Ð6Ñ Its electric potential energy will continuously decrease, but will never quite reach a constant
minimum value.
Š
•A
•B
•C
Questions #7-10 refer to this figure. In this figure, also, a positive charge is fixed in
position at the origin. Suppose a positive charge ; is held at rest at position A, and then
released and allowed to move freely. It passes through position B, and then moves on
toward position C.
7. Which of the following statements about charge ; is true?
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Its kinetic energy is the same at B and A, and its electric potential energy is the same at B and A.
Its kinetic energy is larger at B than at A, and its electric potential energy is larger at B than A.
Its kinetic energy is smaller at B than at A, and its electric potential energy is smaller at B than at A.
Its kinetic energy is larger at B than at A, but its electric potential energy is smaller at B than at A.
Its kinetic energy is smaller at B than at A, but its electric potential energy is higher at B than at A.
8. This question again refers to the situation in Question #7. In comparing the energy of
the charge ; at positions C and B, which of the following statements is true?
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Its kinetic energy is the same at C and B, and its electric potential energy is the same at C and B.
Its kinetic energy is larger at C than at B, and its electric potential energy is larger at C than at B.
Its kinetic energy is smaller at C than at B, and its electric potential energy is smaller at C than at B.
Its kinetic energy is larger at C than at B, but its electric potential energy is smaller at C than at B.
Its kinetic energy is smaller at C than at B, but its electric potential energy is higher at C than at B.
9. Again consider the setup shown in Question #7. Suppose now that a positively charged
particle is shot from a gun that is located far away from the positive charge at the origin,
but which is aimed directly at it. After leaving the gun the particle heads toward the origin,
passing first through position C, then position B, and then position A. In comparing its
energy at positions C and B, which of the following statements is true?
Ð1Ñ
Ð2Ñ
Ð3Ñ
Ð4Ñ
Ð5Ñ
Its kinetic energy and electrical potential energy are both the same at C and B.
Its kinetic energy and electric potential energy are both larger at C than at B.
Its kinetic energy and electric potential energy are both smaller at C than at B.
Its kinetic energy is larger at C than at B, but its electric potential energy is smaller at C than at B.
Its kinetic energy is smaller at C than at B, but its electric potential energy is higher at C than at B.
10. Consider the situation described in #9. Let us call the magnitude of the change in
kinetic energy | ?OI |, and the magnitude of the change in the electric potential energy
|?T I |. Which of these is true about the energy of the particle shot from the gun, as it
travels from position C to position B?
Ð1Ñ | ?OI | = | ?T I |
Ð2Ñ | ?OI | > | ?T I |
Ð4Ñ Not enough information to answer.
Ð3Ñ | ?OI | > | ?T I |
11. In this figure (as in the previous two ones), a positive charge is fixed in position at the
origin. The dotted lines are circles centered at the origin. Suppose a positive charge ; is
placed at rest at position A on the inner circle, and then allowed to move freely until it
passes position B on the outer circle. Its kinetic energy at that point is ÐOIÑB and its
electric potential energy is ÐT IÑB . Suppose now that the charge ; is placed instead at rest
at position C on the inner circle, and allowed to move freely. As it passes position D on the
outer circle, its kinetic energy at that point is ÐOIÑD , and its electric potential energy is
ÐT IÑD . Which of the statements A through E is true?
(1) ÐOIÑB œ ÐOIÑD and ÐT IÑB œ ÐT IÑD
(2) ÐOIÑB œ ÐOIÑD and ÐT IÑB  ÐT IÑD
(3) ÐOIÑB œ ÐOIÑD and ÐT IÑB  ÐT IÑD
(4) ÐOIÑB  ÐOIÑD and ÐT IÑB  ÐT IÑD
(5)
ÐOIÑB  ÐOIÑD and ÐT IÑB  ÐT IÑD
(6) There is not enough information to answer this
question.
Š
•A
•B
•C
It can be shown that the electric potential energy of a charge ; in the neighborhood of
another charge U depends only on the values of ; and U, and on the distance between the two
charges. Use this fact to help answer the questions below. In the diagram, the positive charge
is taken to be at the origin. Suppose that the points A, B, and C in the diagram above are
equally spaced, say, 2 m, 3 m, and 4 m from the positive charge, respectively.
12. A positive charged particle is shot from a gun toward the positive charge at the origin,
starting at the far right, so that it travels on a path that takes it first through position C,
and then through position B. First, the particle is shot from the gun at a "slow" speed, and
the change in its potential energy between points C and B is Ð?T IÑslow . Then the same
particle is shot from the gun at a "fast" speed, and the change in its potential energy
between points C and B is Ð?T IÑfast . Which of the statements shown is true?
(1) Ð?T IÑfast  Ð?T IÑslow
(2) Ð?T IÑfast œ Ð?T IÑslow
(3) Ð?T IÑfast  Ð?T IÑslow
(4) Not enough information to determine how they are related.
13. With regard to the particle discussed in problem #12, which of these statements
regarding its change in the kinetic energy is true?
(1) Ð?OIÑfast  Ð?OIÑslow
(2) Ð?OIÑfast œ Ð?OIÑslow
(3) Ð?OIÑfast  Ð?OIÑslow
(4) Not enough information to determine.
14. With regard to the particle discussed in problem #12, at which point will it have the
largest kinetic energy? (Answer 1 for A, 2 for B, or 3 for C.)
15. With regard to the particle discussed in problem #12, at which point will it have the
smallest kinetic energy? (Answer 1 for A, 2 for B, or 3 for C.)
16. Between what two points will the particle of problem #12 have the greatest change in
kinetic energy? (Consider only the absolute value of the change in kinetic energy.)
(1) Between A and B
(2) Between B and C
(3) No difference
17. What will the particle of problem #12 do?
(1) It will speed up and collide with the positive charge at the origin.
(2) It will slow down but eventually collide with the positive charge at the origin.
(3) It will slow down and then eventually turn around and fly outwards.
Topic 2D. Electric Potential
Electric Potential
We now go from our discussion of electric potential energy to a discussion of electric
potential. What is the difference between electric potential energy and electric "potential"?
We have seen that a charged particle may have different values of electric potential
energy, depending on its location. As it accelerates under the influence of the electric field the
charge's kinetic energy increases, and so its potential energy must be decreasing (since its total
energy remains constant). So, when it is released from rest, a test charge moves toward a
location where its potential energy is lower. If we specify a reference point where the potential
energy is zero, we may "map out" the potential energy value of the test charge at every possible
location.
An example is a test charge ;1 in the neighborhood of an isolated source charge U. If ;1 is a
distance < from U its potential energy is 5;1 UÎ<, as described earlier.
Suppose we draw a circle of radius <a
around U, and suppose ;1 is located somewhere
on that circle. What would happen to the
potential energy of ;1 as it moves around that
circle? Answer: Its T I will remain constant.
Let's suppose that the T I of ;1 at a point
anywhere on this circle is equal to 20 J.
Now suppose we draw another circle, this one at
a radius <b œ 2<a , and we put charge ;1 on that
circle. Would its T I be greater than, less than,
or the same as it was on the smaller circle?
Answer: Less than:
[T I (<b ) œ ½ T I (<a ) œ 10 J.]
We are beginning to "map out" the potential energy of test charge ;1 at various points in
the neighborhood of source charge U. We can see that this map will consist of a set of circles of
different radii; at any point on a given circle, the T I of ;1 will have one specific value. As ;1
moves out to larger circles, its T I decreases. We could refer to various locations on this map as
"the 20 J circle," "the 10 J circle," "between the 20 J circle and the 10 J circle," and so on.
However, there is a drawback to this map: the numerical values depend on which test
charge we use! Suppose we put a charge ;2 = 4;1 on the "20 J" circle. Would its T I be greater
than, less than, or equal to 20 J? The answer is: Greater than, because [T IÐ;2 Ñ œ 4T I (;1 ) = 80
J.] Our map would be more useful if it were completely independent of which test charge we
used. For this reason, we will define a new quantity called electric "potential" [symbol: Z ] (not
the same as "potential energy"); a map (or diagram) of "potential" will allow us to determine the
potential energy of any test charge at any point in the region.
Electric Potential Z is defined as "potential energy per charge," or as the ratio:
Ê
Z œ
ÐT IÑ;
;
Since the PE of a charge depends on where it is, the electric potential Z depends on
t , meaning "the value of Z at the point <t.
position. We sometimes indicate this by writing Z Ð<Ñ
In our example, the inner circle was a "20 J" circle for charge ;1 , and an "80 J" circle for
charge ;# . (This was because T I œ 5;UÎ<, and ;2 œ 4;1 .) Let's suppose that charge ;1 œ 4 C,
and charge ;2 œ 16 C. If we now calculate the potential Z on the inner circle using ;1 , we get
Z œ 20 JÎÐ4 CÑ = 5 J/C; using ;2 , we get Z œ 80 JÎÐ16 CÑ œ 5 J/C. So we calculate Z œ 5 J/C
no matter which test charge we use.
The unit of potential is called the volt (V) [1 V = 1 J/C] and so we may now identify the
inner circle as the "5 V" potential circle, and the outer circle as the "2.5 V" potential circle:
There are two very useful relationships (really just one relationship, written two different
ways) that follow directly from the definition of the potential. They relate the "potential
difference" (?Z )AB ´ ZB  ZA between two points A and B, to the change in potential energy
of a charge q as it moves from point A to point B:
(?Z )AB œ
Ð?T IÑEpF
;
Ð?T IÑApB œ ;Ð?Z ÑAB
We can now answer a question such as this: If a 3-C charge moves from the 5-V circle to
the 2.5-V circle, what will happen to its kinetic energy? Answer: Its T I will decrease by an
amount equal to ; ?Z œ ;ÐZ0  Z3 Ñ œ (3 C)(2.5 V  5 V) œ (3C)(  2.5V) =  7.5 J;
therefore, its OI must increase by  7.5 J. Note that the sign of ?Z is important, because it
determines the sign of ?T I and thus the sign of ?OI .
The word "voltage" is often used as shorthand for "potential difference" (?Z ) . When
you hear "voltage," you should always think of the difference in electric potential between two
different points. There is no such thing as an absolute potential, only the potential at a point
relative to some other point defined to have zero potential. We often loosely use the term
"potential difference" to refer to the absolute value of potential difference, or l?Z l.
We can now map out a region where an electric field is present by representing electric
potential, instead of (or in addition to) drawing vectors to indicate the electric field at various
points. This is normally done by drawing a whole set of lines called "equipotential lines." An
"equipotential line" is a line along which the electric potential does not change. Our circles in the
diagrams above are equipotential lines, since the potential has a constant value along any given
circle. If a charged particle moves along an equipotential line, its potential energy is not
changing. Of course, when we consider three dimensions we realize that equipotential lines are
just lines on a more general equipotential surface.
We can look at a "potential diagram" and immediately determine certain features about
the magnitude and direction of the electric field that is present in the region. For instance:
Consider the electric field vectors in the region between two equipotential circles. Will they
point toward the higher-potential circle, toward the lower-potential circle, or in a direction
tangent to both circles? Answer: They will point from the higher-potential circle toward the
lower-potential circle. Explain how you can determine this by considering the motion of a test
charge; consider both positive and negative test charges.
We can also see here an example of a general relationship: The electric field vector at a
point in space always points perpendicular to the equipotential line that passes through
that point.
These diagrams show the field lines (solid) and equipotential lines (dashed) for cases of
(a) a constant electric field, (b) in the vicinity of a single positive charge, and (c) in the vicinity
of an electric dipole (a negative and a positive charge of the same magnitude).
It is also possible to determine the magnitude of the electric field at various points on a
potential diagram. There are two relationships that will help us do this:
1) In a region where the electric potential is constant (uniform), the electric field is
zero. Suppose a charge is drifting in a region where the electric potential has the same value
everywhere. That means that the charge has the same value of potential energy at every point,
and so its kinetic energy can not be changing, either. Therefore, the charge's velocity is constant
and unchanging as well. We must conclude that no force is acting on this charge to accelerate it,
and so the electric field must be zero in this region. By a similar argument, we can see that if the
electric field is zero in some region, the electric potential must have a uniform, unchanging value
throughout that region.
2) When comparing two regions of equal size, the region with the largest variation
of electric potential will have the electric field with the largest magnitude. Here's an
example: Let's consider two regions of the same size, as shown in the diagram, each of which has
a uniform electric field present. Suppose we put a  3 C charge at point P in region A, and let it
move freely. We know that the electric field points from higher potential to lower potential,
and so the charge will accelerate to the right. When it approaches the right side, its kinetic
energy will have increased by an amount equal to its potential energy loss, so
?OI œ  ?T I œ  ; ?Z œ  Ð3 CÑÐ1 V  4 VÑ œ  Ð3 CÑÐ  3 VÑ œ 9 J.
If we do the same thing in region B, we can see that when the charge has covered about the same
amount of distance as before, the change in its kinetic energy is now  (3 C)(  5 V) = 15 J,
which means its kinetic energy has increased by 15 J.
Let's see what this implies about the electric field. Since ?OI œ [ , more work was
done by the electric force in region B. Since in this situation [ œ J . and . is the same in both
cases, the force J must be greater in region B. Since I œ J Î; and we used the same ; in both
regions, the electric field magnitude I must be greater in region B. Qualitatively, we can
associate more "tightly packed" equipotential lines (as in region B) with larger-magnitude
electric fields. In the case of a uniform electric field (as in A and B), there is a simple
relationship:
I œ Ð?Z ÑÎ. .
The units of the electric field I were originally introduced as N/C, but they are also V/m.
There are a two other important facts related to electric potential:
(1) Every point in a single piece of conducting material has the same value of
electric potential. We say that the conductor is an "equipotential volume." A conductor is a
material in which electric charges can move freely from place to place. (Actually, it is the
negative charges - the electrons - which move, while the positive charges stay fixed. However,
we will usually speak as if it is the positive charges that are really moving. This is a conventional
choice, which helps to simplify the discussion. It is usually irrelevant what the sign of the
moving charges is.) In an "ideal" conductor, there is no electric field present. The charges always
arrange themselves in such a way that the net electric field in the conductor is zero. According to
#1 above, this means that the value of the potential does not change anywhere inside the
conductor. Any piece of conducting material  such as a wire  in direct contact with another
piece of conducting material  such as a piece of metal  will form one single conducting
volume, which will all be at the same electric potential.
(2) A battery is an electrochemical device the supplies electrical energy, and has
two "ends" which are called the positive and negative "terminals." The symbol for a battery
consists of two parallel lines, in which the longer line corresponds to the positive terminal.
Sometimes the terminals are denoted "  " and "  " as well. The battery uses chemical
reactions to ensure that the positive terminal is at a higher electric potential than the negative
terminal. Moreover  and this is very important  in an ideal battery, the potential difference
between the terminals always has the same value. Here are symbols used for batteries.


The battery on the left has a positive terminal on the left and a negative terminal on the right;
the battery on the right has a negative terminal on the left and a positive terminal on the right.
Capacitance
A "capacitor" is a device that is used to store electrical charge and electrical energy.
(When it stores charge, it also stores energy.) It is very different from a battery, because it can be
"emptied" of all of its charge and energy in a very short time period  usually less than one
second. We say that the capacitor is "discharged" when it loses its charge, and that it becomes
"charged" when it acquires its charge and its energy. Capacitors of different types are used in
thousands of applications, including cardiac defibrillators, electric circuits for radios, electronic
flash lamps, computer memory chips, and nearly all electronic devices.
Capacitors come in many varieties of shapes and sizes, but fundamentally they all consist
of one thing: a single pair of conducting objects, fixed in positions close to each other (but not
touching). Usually, the two conducting objects are identical to each other, and often they are
shaped like plates or sheets. (Sometimes the sheets are flexible and are rolled up to form a
compact cylinder.) Perhaps the simplest form of capacitor is a single pair of parallel metal plates,
where the distance between the plates is relatively small compared to the size of the plate. If we
want to "charge" this capacitor, we might connect it to a battery (one plate to the positive
terminal, the other plate to the negative terminal). Then one plate becomes positively charged,
with charge  U, while the other plate acquires negative charge  U. This particular
arrangement is very useful, because it can be shown that the electric field in the region between
the plates is very nearly uniform. In fact, we will always assume that the field between the plates
is completely uniform. As we have seen, in this case the equipotential lines form a set of equallyspaced parallel lines.
The quantity of charge that is stored on the plates depends on the potential difference
?Z between the positive plate (which is at higher potential) and the negative plate Ð?Z œ Z 
Z Ñ. A larger ?Z leads to a larger U. In fact, these quantities are proportional to each other:
Ê
U œ G ?Z ,
where G is a proportionality constant. (Why is this? A large amount of charge on the plates
implies that the electric field between the plates will have a large magnitude  and so a charged
particle moving from one plate to the other will undergo a large change in kinetic (and potential)
energy. This means that the potential difference ?Z would also be large.)
The proportionality constant G is called the "capacitance" of the capacitor, and it
depends only on the size, shape, and spacing of the conducting plates. It will be the same for any
value of U or ?Z . The unit of capacitance is called the farad [symbol: F] so 1 F = 1 C/V.
The amount of energy that is stored by the capacitor depends both on the capacitance,
and on the potential difference:
Ê
Ê
Ê
Energy stored œ
alternatively:
Energy stored œ
alternatively:
Energy stored œ
½ G Ð?Z Ñ2 .
½ U2 ÎG .
½ U ?Z
This is the energy provided to the capacitor when it is charged and provided by the
capacitor when it dischargesÞ The alternate formulas are derived using U œ G ?Z .
Where does this formula come from? The easiest way to understand it is to look at the
last of the three formulas. To "push" a small charge ; onto the capacitor through a potential
difference Z requires energy ;Z . As more and more charge is built up on the capacitor, Z gets
larger and larger. On average, the potential difference through which charge is pushed is half the
final potential difference, or "# Ð?Z Ñ, so the total energy required is UÐ "# ÑÐ?Z Ñ or ½ U ?Z .
The capacitance G of a capacitor depends on the size and shape of the capacitor and the
material in the space between the plates. There is a simple formula for the capacitance of a
parallel-plate capacitor with air between the plates: G œ EÎ415. , where E is the area of each
plate and . is the distance between the plates.
Example 1. A 27-C charge is fixed at the origin. When a 3-C charge is placed at point P, it
has a potential energy of 9 J. If the 3-C charge is now removed, what will be the electric
potential experienced by a 6-C charge placed at point P?
Answer: Z œ T IÎ; , so V = Ð9 JÑÎÐ3 CÑ = 3 V. This will be the potential experienced by
any charge at point P.
Example 2. How much external work must be done to force a  3-C charge to move from
the negative plate to the positive plate of a parallel plate capacitor that is connected to a 6V battery? (Assume that the speed of the charge does not change.)
Answer: [non-cons œ ?X I œ ?OI  ?T I œ 0 +?T I œ ; ?Z œ (3 C)(6 V) = 18 J.
Example 3. Several equipotential lines are shown, with potential values indicated. Where
should you place a proton so that it would experience the largest magnitude of electrical
force?
Answer: Point A; this is where the equipotential lines are most "densely" packed, and
where ?Z Î?B (which is approximately equal to I ) is the greatest.
Example 4. (a) A capacitor is connected to a 5-V battery, which results in a net charge of
+3 C on the positive plate. Suppose the capacitor is disconnected from the 5-V battery, and
connected instead to a 10-V battery. What will be the charge on the negative plate?
Answer: U œ G ?Z ; here, G does not change, but ?Z is doubled. Therefore, the charge
on the positive plate will now be equal to  6 C, while that on the negative plate will be  6 C.
(b) What is the capacitance of this capacitor?
Answer: Starting with U œ G ?Z we can write G œ UÎÐ?Z Ñ œ Ð3 CÑÎÐ5 VÑ œ 0.6 F.
This is a very large capacitance.
Example 5. A 5-.F capacitor is connected to a 20-V battery. What is the charge on each
plate, which results in a net charge of +3 C on the positive plate. Suppose the capacitor is
disconnected from the 5-V battery, and connected instead to a 10-V battery. What will be
the charge on the negative plate?
QUESTIONS ON ELECTRIC POTENTIAL
Prerequisite Concepts:
• Electric potential energy
• Definition of electric potential: Z œ T IÎ;
• Conductor is an equipotential volume
• Electric potential in neighborhood of a point charge U: Z œ 5UÎ<
[Note: All gravitational forces may be ignored in this chapter.]
In this figure, two parallel plates are connected to the positive and negative terminals of a
battery as shown. A 2-C positive charge is held at rest on the left-hand plate, and then
released and allowed to move freely. When it strikes the right-hand plate, its kinetic energy is
12 J. Questions #1-5 refer to this figure.
1. As the 2-C charge moved between the plates, how did its electric potential energy
change?
(1) increased by 6 joules
(2) increased by 12 joules
(3) decreased by 6 joules
(4) decreased by 12 joules
(5) remained constant
(6) not enough information to answer.
2. What is the potential difference between the terminals of the battery?
(1) 0 volts
(2) 3 volts
(3) 6 volts
(6) not enough information to answer.
(4) 12 volts
(5) 24 volts
3. Assume the negative terminal of the battery is at a potential of 0 volts. What is the
electric potential midway between the plates?
(1) 0 volts
(2) 3 volts
(3) 6 volts
(6) not enough information to answer.
(4) 12 volts
(5) 24 volts
4. Suppose a different positively charged particle, starting from rest on the left-hand plate,
moves across and strikes the right-hand plate with a kinetic energy of 24 joules. What is
the charge on this particle?
(1) 1 C
(2) 2 C
(3) 3 C
(4) 4 C
(5) 6 C
(6) 12 C
5. What is the electric potential energy of this second charge when it is midway between
the plates?
(A) 3 J
(B) 6 J
(C) 12 J
(D) 18 J
(E) 24 J
(F) 48 J
In this figure, two parallel metal plates are connected to a 4-V battery as shown. (The vertical
lines divide the space between the plates into four equal segments.) The electric potential
point #1 is 0 volts. Choose your answers for questions #6-10 from this list:
(1) 0 V
(2) 1 V
(3) 2 V
(4) 3 V
(5) 4 V
(6) none of the above
6. What is the electric potential at point #2?
7. What is the electric potential at point #4?
8. What is the electric potential at point #5?
9. What is the electric potential at point #6?
10. What is the electric potential at point #3?
In this figure, a metal strip (points A and B are located on this strip) and a metal disc are
connected to the terminals of a ten-volt battery as shown. The lines that are drawn are
equipotential lines, drawn at one-volt intervals. The electric potential at point A is 0 volts.
11. The electric potential at a point on the metal disc is
(1) 0 V
(2) 2 V
(3) 4 V
(4) 6 V
(5) 8 V
(6) 10 V
(5) 8 V
(6) 10 V
12. What is the electric potential at point B?
(1) 0 V
(2) 2 V
(3) 4 V
(4) 6 V
13. What is the electric potential at point C?
(1) 1 V
(2) 3 V
(3) 5 V
(4) 7 V
(5) 9 V
14. What is the difference in electric potential between points D and C?
(1) 1 V
(2) 2 V
(3) 3 V
(4) 4 V
(5) 5 V
(6) 6 V
15. At which point does the electric potential vary most rapidly as a function of distance?
In this figure, two parallel metal plates are connected to the terminals of a battery as shown.
The vertical lines divide the space between the plates into four equal segments. Questions #612 refer to this figure.
6. Rank, in order, the electric potentials at the points A through F, starting with the largest
value; if two or more are at the same potential, put an "equals" sign ["="] between them.
E.g., A, B=C, D, E, F would mean A is largest, F is smallest, and B is equal to C but is
larger than D.
Ranking: (largest) _________________________________________ (smallest)
7. Rank, in order, the electric field magnitudes at the points A through F, in the same
manner.
Ranking: (largest) _________________________________________ (smallest)
8. If ZA represents the electric potential at A, and so forth, rank the following magnitudes
(which are the absolute values of the quantities indicated):
(1) |ZA  ZC |
(2) |ZD  ZF |
(3) |ZB  ZC |
(4) |ZB  ZE |
(5) |ZB  ZF |
9. Suppose a proton is placed at point A, initially at rest, and allowed to drift out to the line
on which points C and D lie. Then another proton, also initially at rest, is placed at point B
and allowed to drift out to the line on which points E and F lie. A third proton, initially at
rest, is placed at point C, and allowed to drift out to the line on which points E and F lie.
Rank the velocities of the three particles that started at A, B, and C.
Ranking: (largest) _________________________________________ (smallest)
10. Suppose you have to push a proton in (at constant speed) from point E to point A, and
then you have to push another proton in from point C to point B (also at constant speed).
Finally, you have to push another proton is (at constant speed) from point F to point C.
Rank the magnitude of the work you have to do:
(A) The work needed to push a proton from E to A
(B) The work needed to push a proton from C to B
(C) The work needed to push a proton from F to C
Ranking: (largest) _________________________________________ (smallest)
11. If a charge moves at constant velocity from point A to point B, is the amount of
external work required:
(1) positive
(2) negative
(3) zero
12. As a proton drifts from one plate to another, the electrical force on it:
(1) increases
(2) decreases
(3) remains the same