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CHAPTER·
21
Electric Fields
~ High Energy Halo
Research into ultra-high-voltage-up
to two million voltstransmission of electricity is essential for development of
future power technology. At a special test facility, this ultrahigh-voltage test lir::ledemonstrates a man-made corona
effect. Why is the corona visible around this power line
when the effect is not seen in the electric wires in our
homes?
Chapter Outline
21.1 CREATING AND
MEASURING ElECTRIC
FIELDS
· The Electric Field
· Picturing the Electric Field
21.2 APPLICATIONS OF THE
ElECTRIC FIELD
· Energy and the Electric
Potential
· The Electric Potential in a
Uniform Field
· Millikan's Oil Drop Experiment
· Sharing of Charge
· Electric Fields Near
Conductors
· Storing Electric Energy-The
Capacitor
{Concept Check
T
he blue glow around the wires in the photo is a modern
version of "St. Elmo's Fire." Sailors in the time of Columbus saw these ghostly-colored
streamers issuing from their
high ships' masts. They recognized them as warning signs
of an approaching lightning storm. The glow is indeed related to lightning. The man-made "fire"
in the photo is
called a corona discharge. Besides causing the eery glow, it
causes radio interference and can start a spark jumping from
the wire to another conductor, so designers of these experimental power lines try to reduce or eliminate it entirely.
The following terms or concepts
from earlier chapters are
important for a good
understanding of this chapter. If
you are not familiar with them,
you should review them before
studying this chapter.
· gravitational field, Chapter 8
· gravitational potential energy,
Chapter 11
· electrical interactions,
Coulomb's law, Chapter 20
425
ObjeCtives
21.1
• define an electric field; explain how
to measure it.
• distinguish between force and field.
• be able to solve problems relating
field, force, and charge.
• distinguish between electric field
and field lines.
T
Charge and Field
Conventions
• Positive charges are red.
• Negative charges are blue.
~ Electric fields are red.
POCKET
LAB
ELECTRIC FIELDS
Use a 5-cm long thread to
hang a pith ball (or small plasticfoam ball) from the end of a dangling straw. Rub a 25-cm square
piece of plasticfoam with fur to
put a large negative charge on
the foam. Stand the foam in a
vertical orientation.Give the pith
ball a positive test charge (use
the fur). Notice that the pith ball
hangs straight down until you
bring it towardthe chargedfoam.
The pith ball is now in an electric
field. How does the pith ball indicate the "strength" of the field?
Move the test charge (pith ball)
to investigate the electric field
around the foam. Record your
results.
426
Electric Fields
......................................
CREATING AND MEASURING
ELECTRIC FIELDS
he electric force, like the gravitational force, varies inversely as the
square of the distance between two objects. Both forces can act at a
great distance. How can a force be exerted across what seems to be
empty space?In trying to understand the electric force, Michael Faraday
(1791-1867) developed the concept of an electric field. According to
Faraday, a charge creates an electric field about it in all directions. If a
second charge is placed at some point in the field, the second charge
interacts with the field at that point. The force it feels is the result of a
local interaction. Interaction between particles separated by some distance is no longer required.
The Electric Field
It is easy to state that a charge produces an electric field. But how
can the field be detected and measured? We will describe a method
that can be used to measure the field produced by an electric charge q.
You must measure the field at a specific location, for example, point
A. An electrical field can be observed only because it produces forces
on other charges, so a small positive test charge is placed at A. The
force exerted on the test charge, q', at this location is measured. According to Coulomb's law, the force is proportional to the test charge.
If the size of the charge is doubled, the force is doubled. Thus, the ratio
of force to charge is independent of the size of the charge. If we divide
the magnitude of the force, F, on the test charge, measured at point A,
by the size of the test charge, q', we obtain a vector quantity, Flq'. This
quantity does not depend on the test charge, only on the charge q and
the location of point A. We call this vector quantity the magnitude of
the electric field. The electric field at point A, the location of q', is
[E~FO""I q'
The direction of the electric field is the direction of the force on the
positive test charge. The magnitude of the electric field intensity is measured in newtons per coulomb, N/C.
Notice that just as the electric field is the force per unit charge, the
gravitational field is the force per unit mass, g = Flm.
We said that an electric field should be measured by a small test
charge. Why? The test charge exerts a force on q. We want to make
sure the force exerted by the test charge doesn't move q to another
location, and thus change the force on q' and the electric field we are
measuring.
So far we have found the field at a single point. The test charge is
now moved to another location. The force on it is measured again and
the electric field at that location calculated. This process is repeated
again and again until every location in space has a measurement of the
vector quantity, the electric field, on the test charge associated with it.
F. Y. I.
Example Problem
Calculating the Magnitude
of the Electric Field
A positive test charge of 4.0 x 10-5 C is placed in an electric
field. The force on it is 0.60 N acting at 10°. What is the magnitude
and direction of the electric field at the location of the test charge?
Unknown:
Given: size of test charge,
q' = 4.0 X 10-5 C
electric field, E
F
Basic equation:
force on test charge,
F = 0.60 N at 10°
E
Robert Van de Graaff devised
the high-voltage electrostatic
generator in the 1930s. These
generators can build up giant potentials that can accelerate particles to high energies.
= ----;
q
Solution:
Find the size of the field intensity.
E
F
0.60N
= q' = 4.0 x 10-5
C
=
1
.5
x
104N/C
The field direction is in the direction of the force since it is a positive test charge, so
E
=
1.5
X
104 N/C at 10°.
If the force were in the opposite direction, the field at the test
charge would also be in the opposite direction, that is, E = 1.5 X
104 N/C at 10° + 180° = 190°.
Practice Problems
1. A negative charge of 2.0 x 10-8 C experiences a force of 0.060 N
to the right in an electric field. What is the field magnitude and direction?
2. A positive test charge of 5.0 x 10-4 C is in an electric field that
exerts a force of 2.5 x 10-4 N on it. What is the magnitude of the
electric field at the location of the test charge?
3. Suppose the electric field in Practice Problem 2 were caused by a
point charge. The test charge is moved to a distance twice as far from
the charge. What is the magnitude of the force that the field exerts
on the test charge now?
~ 4. You are probing the field of a charge of unknown magnitude and
sign. You first map the field with a 1.0 x 10-6 C test charge, then
repeat your work with a 2.0 x 10-6 C charge.
a. Would you measure the same forces with the two test charges?
Explain.
b. Would you find the same fields? Explain.
The collection of all the force vectors on the test charge is called an
electric field. Any charge placed in an electric field experiences a force
on it due to the electric field at that location. The strength of the force
depends on the magnitude of the field, E, and the size of the charge, q.
Thus F = Eq. The direction of the force depends on the direction of the
field and the sign of the charge.
The total electric field is the vector sum
of the fields of individual charges.
21.1 Creating and Measuring Electric Fields
427
FIGURE 21-1. An electric field can
be shown by using arrows to
represent the direction of the field at
various locations.
Electric field
Field charge
e
A picture of an electric field can be made by using arrows to represent the field vectors at various locations, Figure 21-1. The length of
the arrow shows the magnitude of the field; the direction of the arrow
shows its direction.
To find the field from two charges, the fields from the individual
charges can be added vectorially. Or, a test charge can be used to map
out the field due to any collection of charges. Typical electric fields
produced by charge collections are shown in Table 21-1.
Metal dome
Insulator
Picturing the Electric Field
An alternative picture of an electric field is shown in Figure 21-3.
The lines are called electric field lines. The direction of the field at any
point is the tangent drawn to the field line at that point. The strength of
the electric field is indicated by the spacing between the lines. The field
is strong where the lines are close together. It is weaker where the lines
are spaced farther apart. Remember that electric fields exist in three
dimensions. Our drawings are only two-dimensional models.
The direction of the force on the positive test charge near a positive
charge is away from the charge. Thus, the field lines extend radially outward like the spokes of a wheel, Figure 21-3a. Near a negative charge the
direction of the force on the positive test charge is toward the charge,
so the field lines point radially inward, Figure 21-3b.
Table 21-1
Approximate Values of Typical Electric Fields
FIGURE 21-2. Charge is transferred
onto a moving belt at A, and from
the belt to the metal dome at B. An
electric motor does the work needed
to increase the electric potential
energy.
428
Electric Fields
Field
Near charged hard rubber rod
In television picture tube
Needed to create spark in air
At electron orbit in hydrogen atom
Value (N/C)
1
1
3
5
x
x
X
X
103
105
106
10"
a
c
b
FIGURE 21-3. Lines of force are
drawn perpendicularly away from the
positive object (a) and
perpendicularly into the negative
object (b). Electric field lines
between like- and oppositely-charged
objects are shown in (c).
When there are two or more charges, the field is the vector sum of
the fields due to the individual charges. The field lines become curved
and the pattern complex, Figure 21-3c. Note that field lines always
leave a positive charge and enter a negative charge.
The Van de Graaff machine is a device that transfers large amounts
of charge from one part of the machine to the top metal terminal Figure
21-2. A person touching the terminal is charged electrically. The
charges on the person's hairs repel each other, causing the hairs to follow the field lines. Another method of visualizing field lines is to use
grass seed in an insulating liquid such as mineral oil. The electric forces
cause a separation of charge in each long, thin grass seed. The seeds
then turn so they line up along the direction of the electric field. Therefore, the seeds form a pattern of the electric field lines. The patterns in
Figure 21-4 were made this way.
Field lines do not really exist. They are just a means of providing a
model of an electric field. Electric fields, on the other hand, do exist.
An electric field is produced by one or more charges and is independent
of the existence of the test charge that is used to measure it. The field
provides a method of calculating the force on a charged body. It does
not explain, however, why charged bodies exert forces on each other.
That question is still unanswered.
a
b
c
d
An electric field points away from
positive charges and toward negative
charges.
Electric fields are real. Electric field
lines are imaginary, but help in making
a picture of the field.
FIGURE 21-4. Lines of force
between unlike charges (a, c) and
between like charges (b, d) describe
the behavior of a positively-charged
object in a field. The top
photographs are computer tracings
of electric field lines.
21.1 Creating
and Measuring
Electric
Fields
429
PH Y
SICSLAB :: Voltage
Charges, Energy,
Purpose
To make a model that demonstrates the relationship of charge, energy, and voltage.
4. Tape the 3-V rectangle to the 3" mark on the
ruler, the 6-V to the 6" mark, and so on.
5. Let each steel ball represent 1 C of charge.
6. Lift and tape 4 steel balls to the 3-V rectangle, 3
to the 6-V rectangle, and so on.
Materials
·
·
·
·
ball of clay
ruler
cellophane tape
12 of 3-mm diameter steel balls
Procedure
1. Use the clay to support the ruler vertically on
the tabletop. (The A" end should be at the table.)
2. Cut a 2 cm x 8 cm rectangular piece of paper
and write on it "3 V = 3 J/C".
3. Cut three more rectangles and label them: 6 V =
6 J/C, 9 V = 9 J/C 12 V = 12 JlC
Observations and Data
1. The model shows different amounts of charges
at different energy levels. Where should steel
balls be placed to show a zero energy level? Explain.
2. Make a data table with the columns labeled
"charge", "voltage", and "energy".
3. Fill in the data table for your model for each
level of the model.
Analysis
1. How much energy is required to lift each coulomb of charge from the tabletop to the 9-V
level?
2. What is the total potential energy stored in the
9-V level?
3. The total energy of the charges in the 6-V level
is not 6 J. Explain this.
4. How much energy would be given off if the
charges in the 9-V level fell to the 6-V level?
Explain.
Applications
1. A 9-V battery is very small. A 12-V car battery
is very big. Use your model to help explain why
two 9-V batteries would not start your car.
430
Electric Fields
........................
CONCEPT REVIEW
1.1 Suppose you are asked to measure the electric field in space. Answer the questions of this step-by-step procedure.
a. How do you detect the field at a point?
b. How do you determine the magnitude of the field?
c. How do you choose the size of the test charge?
d. What do you do next?
1.2 Suppose you are given an electric field, but the charges that produce the field are hidden. If all the field lines point into the hidden
region, what can you say about the sign of the charge in that region?
1.3 How does the electric field, E, differ from the force, F, on the test
charge?
1.4 Critical Thinking: Figure 21-4b shows the field from two like
charges. The top positive charge in Figure 21-3c could be considered a test charge measuring the field due to the two negative
charges. Is this positive charge small enough to produce an accurate
measure of the field? Explain.
21.2
APPLICATIONS OF THE
ELECTRIC FIELD
Objectives
T
he concept of energy is extremely useful in mechanics. The law of
conservation of energy allows us to solve problems without knowing
the forces in detail. The same is true in our study of electrical interactions. The work done moving a charged particle in an electric field can
result in the particle gaining either potential or kinetic energy, or both.
In this chapter, since we are investigating charges at rest, we will discuss only changes in potential energy.
Energy and the Electric Potential
Recall the change in potential energy of a ball when you lift it, Figure
21-5. Both the gravitational force, F, and the gravitational field, g =
Flm, point toward Earth. If you lift a ball against the force of gravity,
you do work on it, and thus its potential energy is increased.
a
b
Ball
-\
• define the electric potential
difference in terms of work done
moving a unit test charge;
distinguish potential from potential
difference.
• know the units of potential; solve
problems involving potential in
uniform electric fields.
• understand how Millikan used
electric fields to find the charge of
the electron.
· understand how minimizing energy
determines sharing of charge;
define grounding and relate to
charge sharing.
• know where charges reside on
solid and hollow conductors;
recognize the relationship between
conductor shape and field strength.
· define capacitance; describe the
principle of the capacitor; solve
capacitor problems.
Increase in
gravitational
potential
energy
FIGURE 21-5. Work is needed to
move an object against the force of
gravity (a) and against the electric
force (b). In both cases, the potential
energy of the object is increased.
21.2 Applications of the Electric Field
431
F. Y. I.
"Even if I could be Shakespeare, I think I should still
choose to be Faraday."
-Aldous
Huxley (1925)
The situation is similar with two unlike charges. These two attract
each other, and so you must do work to pull one away from the other.
When you do the work, you store it as potential energy. The larger the
test charge, the greater the increase in its potential energy, LlPE.
The electric field at the location of the test charge does not depend
on the size of the test charge. The field is the force per unit charge, or
Flq'. In the same way, we will define a quantity that is the change in
potential energy per unit charge. This quantity is called the electric potential difference and is defined by the equation Ll V = LlPElq'. The unit
of electric
Electric potential difference is change in
potential energy divided by the charge.
The volt is the unit of electric potential.
The reference point of zero level of
potential is arbitrary.
potential,
joule
per coulomb,
is called
potential energy; you give it a positive change in potential energy. Because the test charge is positive, the electric potential difference
between A and B is also positive. The electric potential at B is larger than
the electric potential at A.
If you return the test charge from B back to A, the electric field does
work on you. The potential energy of the charge is reduced. Therefore,
the electric potential difference between B and A is negative. The work
done on you returning the charge is equal and opposite to the work you
do moving it away. Therefore, the electric potential difference between
A and B is equal and opposite to the difference between Band A. The
net change in potential going from A to B to A is zero. Thus, the potential of point A, or any other point, depends only on its position, not on
the path taken to get there.
potential.
The reference,
ences in electric potential
are
= V).
Consider the situation shown in Figure 21-5. When you move the
test charge from A to B, you do work on it. As a result, you increase its
As is the case for any kind of potential
electric potential energy can be measured.
Only potential differences
meaningful.
a volt (J/C
energy, only differences
in
The same is true of electric
or zero, level is arbitrary. Thus, only differare important. We define the potential differ-
ence from point A to point B to be V = VB - VA' Potential differences
are measured with a voltmeter.
Sometimes the potential difference
is
simply called the voltage.
As described in Chapter 11, the potential energy of a system can be
defined to be zero at any convenient
reference point. In the same way,
the electric potential of any point, for example point A, can be defined
to be zero. If VA = 0, then VB = V. If instead, VB = 0, then VA = - V.
No matter what reference point is chosen, the value of the potential
difference from point A to point B will always be the same.
a
Low V
F
F
High V
b
__--I
FIGURE 21-6. Electric potential
energy is smaller when two unlike
charges are closer together (a) and
larger when two like charges are
closer together (b).
432
Electric Fields
+
High V
Low V
FIGURE 21-7. An electric field
between parallel plates.
We have seen that electric potential increases as a positive test charge
is separated from a negative charge. What happens when a positive test
charge is separated from a positive charge? There is a repulsive force
between these two charges. Potential energy decreases as the two
charges are moved farther apart. Therefore, the electric potential is
smaller at points farther from the positive charge, Figure 21-6.
The Electric Potential in a Uniform Field
How does the potential difference depend on the electric field? In the
case of the gravitational field, near the surface of Earth the gravitational
force and field are relatively constant. A constant electric force and field
can be made by placing two large flat conducting plates parallel to each
other. One is charged positively and the other negatively. The electric
field between the plates is constant except at the edges of the plates. Its
direction is from the positive to the negative plate. Figure 21-7 shows
grass seeds representing the field between parallel plates.
In a constant gravitational field, the change in potential energy when
a body of mass m is raised a distance h is given by LlPE = mgh. Remember that g, the gravitational field intensity near Earth, is given by
g = Flm. The gravitational potential, or potential energy per unit of
mass, is mghlm = gh.
What is the potential difference between two points in a uniform electric field? If a positive test charge is moved a distance d, in the direction
opposite the electric field direction, the change in potential energy is
given by LlPE = + Fd. Thus, the potential difference, the change in
potential energy per unit charge, is V = + Fdlq = + (Flq)d. Now, the
electric field intensity is the force per unit charge, E = Flq. Therefore,
the potential difference, V, between two points a distance d apart in a
uniform field, E, is given by
V
=
The electric field between two parallel
plates is uniform.
Ed.
The potential increases in the direction opposite the electric field
direction. That is, the potential is higher near the positively-charged
plate. By dimensional analysis, the product of the units of E and d is
(N/C) . m. This is equivalent to a l/C, the definition of a volt.
21 .2 Appl ications of the Electric Field
433
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If you have a knack for fixing
things, and can learn quickly from
our in-house training on specific
equipment, we would like to hear
from you. You must be able to read
electrical diagrams and manuals and
use sophisticated test equipment.
Vocational school training is a plus,
but experience will be considered.
We need reliable, bondable, productive people willing to work when
needed and interested in advancement. F information contact:
ational Automatic Merchan ism
Association, 20 North Wacker Drive,
Chicago, IL 60606
Example Problem
Potential Difference
Between Two Parallel Plates
Two large, charged parallel plates are 4.0 cm apart. The magnitude of the electric field between them is 625 N/C. a. What is the
potential difference between the plates? b. What work is done moving a charge equal to that of one electron from one plate to another?
Given: d = 0.040 m
E
=
Solution: a. V
b. W
=
=
V, W
Unknowns:
625 N/C
Basic equations:
Ed
(625 N/C)(0.040
m)
25 N . mlC = 25 J/C
25 V
qV
(1.6 X 10-19 C)(25 V)
4.0 X 10-18 CV = 4.0
X
V
W
10-18
=
=
Ed
qV
J
Example Problem
Electric Field Between Two Parallel Plates
A voltmeter measures the potential difference between two large
parallel plates to be 60.0 V. The plates are 3.0 cm apart. What is the
magnitude of the electric field between them?
Unknown: E
Basic equation:
Given: V = 60.0 V
d
=
0.030 m
Solution:
V
=
Ed, so E
V
=
Ed
V
= d
60.0 V
0.030 m
60.0 J/C
0.030 m
2.0 x 103 N/C
Practice Problems
5. The electric field intensity between two large, charged, parallel metal
plates is 8000 N/C. The plates are 0.05 m apart. What is the potential
difference between them?
6. A voltmeter reads 500 V when placed across two charged, parallel
plates. The plates are 0.020 m apart. What is the electric field between them?
7. What potential difference is applied to two metal plates 0.500 m
apart if the electric field between them is 2.50 x 103 N/C?
~ 8. What work is done when 5.0 C is raised in potential by 1.5 V?
434
Electric Fields
P
hysicsand
society
MINIMOTORS
W
hat miracles could a doctor perform with a pump
or motor small enough to fit
into a blood vessel? Drugs, for
example, could be administered precisely where needed in
the exact amount required.
Tiny rotating knives could remove plaque from arteries.
The parts for such machines
could not be made by normal
methods. The new techniques
are often called nanotechnology methods. For more than ten
years, scientists have adapted
the methods used to make computer chips in order to produce
electric motors and gears the
diameter of a human hair. At
this time, however, the most
successful products are sensors.
These sensors detect gas or Iiquid pressure and produce an
electrical signal. They are already used to improve the efficiency of an auto engine and to
monitor blood pressure inside a
human. Other sensors can detect the presence of minuscule
amounts of poisonous gases.
Sensors still in development
measure acceleration more sensitively than any existing device.
Nanotechnology
can also
produce actuators. An actuator
converts an electrical signal
into a mechanical force. One
device, the size of a postage
stamp, moves two million tiny
mirros to project a computer
display on the wall. One motor
that was built uses electric
fields to cause the rotor to spin.
The magnetic forces that run ordinary motors have also been
used to spin micromotors.
The rotors of early motors
were as flat as pancakes and
couldn't do any real work. Using a technique called L1GA,
developed in Germany, scientists at the University of Wisconsin at Madison
created
metal gears one hundred times
thicker than the rotors of the
early motors. Electric charges
were used to assemble these
gears into six-gear transmissions.
DEVELOPING A
VIEWPOINT
The development of nanotechnology is expensive and
may not result in products that
can be sold for decades, if ever.
Who should finance the development? Read further and come
to class prepared to discuss the
following questions.
1. What appl ications do you
foresee in some of the following
fields: medicine, health and nutrition, transportation, and consumer electron ics?
2. You are a scientist who
wants to develop an application
of nanotechnology in health,
but you need money to support
your project. You go to a government agency to ask for
funds. What argument might
you use to convince them to
use tax money to support your
work?
SUGGESTED READINGS
Heppenheimer, T. A. "Microbots." Discover, March 1989,
pp. 78-84.
Gannon,
Robert. "Micromachine Magic./I Popular Science,
March 1989, pp. 88-92.
Stewart, Doug. "New Machines
Are Smaller Than a Hair, and
Do Real Work./I Smithsonian,
October 1990, pp. 85-95.
Hapgood, Fred. "No Assembly
Required." Omni, May 1990,
pp. 66-70.
21.2 Applications of the Electric Field
435
FIGURE 21-8. This illustration shows
a cross-sectional view of Millikan's
apparatus for determining the charge
on an electron.
Charged
Charged
plate
plate
Millikan's Oil Drop Experiment
Millikan measured the charge of an
electron.
A drop is suspended if the electric and
gravitational forces are balanced.
POCKET
LAB
UNIT CHARGE
Place a small container on a
balance and determine the mass
of the empty container. (On an
electronic balance, push the tare
bar.) Place a small handful of
3-mm diameter steel balls in the
container. Don't count them!
Record the mass. Dump the
balls out of the container and repeat the procedure with a different number of balls. Don't count!
Record the mass. Repeat this
loading and massing process a
third time. Look closely at your
mass values. Suggest a possible
unit mass of each ball. Could
your unit mass be too large?
Could it be too small? Explain.
436
Electric Fields
One important application of the uniform electric field between two
parallel plates was the measurement of the charge of an electron. This
was made by American physicist Robert A. Millikan (1868-1953) in
1909.
Figure 21-8 shows the method used by Millikan to measure the
charge carried by a single electron. Fine oil drops were sprayed from
an atomizer into the air. These drops were often charged by friction
with the atomizer as they were sprayed. Gravity acting on the drops
caused them to fall. A few entered the hole in the top plate of the apparatus. A potential difference was placed across the two plates. The
resulting electric field between the plates exerted a force on the charged
drops. When the top plate was made positive enough, the electric force
caused negatively-charged drops to rise. The potential difference between the plates was adjusted to suspend a charged drop between the
plates. At this point, the downward force of the weight and the upward
force of the electric field were equal in magnitude.
The magnitude of the electric field, E, was determined from the potential difference between the plates. A second measurement had to be
made to find the weight of the drop, mg, which was too tiny to measure
by ordinary methods. To make this measurement, a drop was first suspended. Then the electric field was turned off and the rate of the fall of
the drop measured. Because of friction with the air molecules, the oil
drop quickly reached terminal velocity. This velocity was related to the
mass of the drop by a complex equation. Using the measured terminal
velocity to calculate mg, and knowing E, the charge q could be calculated. Millikan found that the drops had a large variety of charges.
When he used X rays to ionize the air and add or remove electrons
from the drops, he noted, however, that the changes in the charge were
always a multiple of -1.6 x 10-19 C. The changes were caused by
one or more electrons being added to or removed from the drops. He
concluded that the smallest change in charge that could occur was the
amount of charge of one electron. Therefore, Millikan said that each
electron always carried the same charge, -1.6 x 10-19 C. Millikan's
experiment
showed
that charge is quantized.
can have only a charge with a magnitude
of the charge of the electron.
The presently
accepted
theory
This means that an object
that is some integral
multiple
says that protons
are made
of matter
up of fundamental
particles called quarks. The charge on a quark is
either + 1/3 or - 2/3 the charge on an electron. A theory of quarks that
agrees with other experiments states that quarks can never be isolated.
Many experimenters
have used an updated Millikan
apparatus to look
for fractional charges on drops or tiny metal spheres. There have been
no reproducible
discoveries
of fractional
charges. Thus, no isolated
quark has been discovered;
the quark theory remains consistent with
experiments.
Example Problem
Finding the Charge on an Oil Drop
An oil drop weighs 1.9 x 10-14 N. It is suspended in an electric
field of intensity 4.0 X 104 N/C.
a. What is the charge on the oil
drop?
b. If the upper plate is positive, how many excess electrons
does it have?
Given: W
£
Solution:
=
mg
1.9 x 10-14 N
=
4.0
=
x 104
a. Electronic
mg
us, q
=
The application of an electric
stimulus causes a series of
changes in the resting membrane potential (RMP) of nerve
and muscle cells. A stimulus that
is applied at one point on a cell
causes a small region of the cell
membrane to become depolarized. This is called a local potential. If large enough, the local potential can cause an action
potential, which is a larger depolarization of the RMP. This
spreads without changing
its
magnitude over the entire cell
membrane.
The drop's mass was found from its
terminal velocity.
q
N/C
and gravitational
h
T
Unknown: excess charge,
Basic equation: £q = mg
. .. .
.
BIOLOGY
CONNECTION
E
forces balance,
so £q
mg.
14
1.9 x 10N
= 4.0 X 104 N/C
= 4.8
x 10-19
The charge of an electron is -1.6
10 19 C.
x
C
(total charge on drop)
b. n u m be r of electro n s = -'--,----"'----,------'-'(charge per electron)
4.8 x 10-19 C
1.6
x 10
19
Cle
3 electrons
There are three extra electrons because a negatively-charged
is attracted toward a positively-charged
plate.
drop
Practice Problems
9. A drop is falling
field is off.
in a Millikan
oil drop apparatus
when
the electric
a. What are the forces on it, regardless of its acceleration?
b. If it is falling
forces on it?
at constant
10. An oil drop weighs 1.9
field of 6.0 x 103 N/C.
a. What
b. How
velocity,
x 10-15
what
can be said about
N. It is suspended
the
in an electric
is the charge on the drop?
many excess electrons
does it carry?
21.2 Applications of the Electric Field
437
11. A positively-charged oil drop weighs 6.4 x 10-13 N. An electric
field of 4.0 x 106 N/C suspends the drop.
a. What
b. How
~ 12. If three
Problem
is the charge on the drop?
many electrons is the drop missing?
more electrons were removed from the drop in Practice
11, what field would be needed to balance the drop?
Sharing of Charge
Charges move until all parts of a
conductor are at the same potential.
If a large and a small sphere have the
same charge, the large sphere will have
a lower potential.
All systems, mechanical and electrical, come to equilibrium when the
energy of the system is at a minimum. For example, if a ball is put on
a hill, it will finally come to rest in a valley where its gravitational potential energy is least. This same principle explains what happens when
an insulated, negatively-charged metal sphere, Figure 21-9, touches a
second, uncharged sphere.
The excess negative charges on sphere A repel each other. Thus the
potential of sphere A is high. We can choose the potential of the neutral
sphere, B to be zero. When one charge is transferred from sphere A to
sphere B, the potential of sphere A is reduced because it has fewer
excess charges. The potential of sphere B does not change because no
work is done adding the first extra charge to this neutral sphere. As
more charges are transferred, however, work must be done on the
charge to overcome the growing repulsive force between it and the
other charges on sphere B. Therefore, the potential of sphere B increases as the potential of sphere A decreases. Negative charges continue to flow from sphere A to sphere B until the work done adding a
charge to sphere B is equal to the work gained in removing a charge
from sphere A. The potential of sphere A now equals the potential of
sphere B. Thus charges flow until all parts of a conducting body, the
two touching spheres in this case, are at the same potential.
Consider a large sphere and a small sphere that have the same
charge, Figure 21-10. The larger sphere has a larger surface area, so
charges can spread farther apart than they can on the smaller sphere.
With the charges farther apart, the repulsive force between them is reduced. Therefore, as long as the spheres have the same charge, the
Metal spheres of equal size
Charged sphere
FIGURE 21-9. A charged sphere
shares charge equally with a neutral
sphere of equal size.
438
Electric Fields
-,
a
b
Metal spheres
Low V
(b)
High V
Same
q
of unequal
size
FIGURE 21-10. A charged sphere
gives much of its charge to a larger
sphere.
Same V
different
q
potential on the larger sphere is lower than the potential on the smaller
sphere. If the two spheresare now touched together, charges will move
to the sphere with the lower potential; that is, from the smaller to the
larger sphere. The result is a greater charge on the larger sphere when
two different-sized spheres are at the same potential.
Earth is a very large sphere. If a charged body is touched to Earth,
almost any amount of charge can flow without changing Earth's potential. When all the excess charge on the body flows to Earth, the body
becomes neutral. Touching an object to Earth to eliminate excess
charge is called grounding. Moving gasoline trucks can become
charged by friction. If that charge were to jump to Earth through gasoline vapor, it could cause an explosion. Instead, a metal wire safely
conducts the charge to ground, Figure 21-11. If a computer or other
sensitive instrument were not grounded, static charges could accumulate, raising the potential of the computer. A person touching the computer could suddenly lower the potential of the computer. The charges
flowing through the computer to the person could damage the equipment or hurt the person.
The charges are closer together at
sharp points of a conductor. Therefore,
the field lines are closer together; the
field is stronger, Figure 21-10.
If the two spheres are at the same
potential, the large one will have
greater charge.
FIGURE 21-11.
fuel truck.
Ground wire on a
21.2 Applications of the Electric Field
439
Electric Fields Near Conductors
All charges are on the outside of a
conductor.
Electric fields are largest near sharp
points.
F. Y. I.
Even if the inner surface is pitted or bumpy, giving it a larger
surface area than the outer surface, the charge will still be entirely on the outside.
The charges on a conductor are spread as far apart as they can be to
make the energy of the system as low as possible. The result is that all
charges are on the surface of a solid conductor. If the conductor is hollow, excess charges will move to the outer surface. If a closed metal
container is charged, there will be no charges on the inside surfaces of
the container. In this way, a closed metal container shields the inside
from electric fields. For example, people inside a car are protected from
the electric fields generated by lightning. On an open coffee can there
will be very few charges inside, and none near the bottom.
Even though the electric potential is the same at every point of a
conductor, the electric field around the outside of it depends on the
shape of the body as well as its potential. The charges are closer together at sharp points of a conductor. Therefore, the field lines are
closer together; the field is stronger, Figure 21-12. The field there can
become so strong that nearby air molecules are separated into electrons
and positive ions. As the electrons and ions recombine, energy is released and light is produced. The result is the blue glow of a corona.
The electrons and ions are accelerated by the field. If the field is strong
enough, when the particles hit other molecules they will produce more
ions and electrons. The stream of ions and electrons that results is a
plasma, a conductor. The result is a spark, or, in extreme cases, lightning. To reduce corona and sparking, conductors that are highly
charged or operate at high potentials are made smooth in shape to reduce the electric fields.
On the other hand, lightning rods, Figure 21-13, are pointed so that
the electric field will be strong near the end of the rod. Air molecules
are pulled apart near the rod, forming the start of a conducting path
from the rod to the clouds. As a result of the sharply-pointed shape,
charges in the clouds spark to the rod rather than to a chimney or other
high point on a house. From the rod, a conductor takes the charges
safely to the ground.
Storing Electric Energy-The
Capacitor
When you lift a book, you increase its potential energy. This can be
interpreted as "storing" energy in a gravitational field. In a similar way,
you can store energy in an electric field. In 1746, the Dutch physician
and physicist Pieter Van Musschenbroek invented a device that could
a
FIGURE 21-12. The electric field
around a conducting body depends
on the structure and shape (b is
hollow) of the body.
440
Electric
Fields
(c)
b
c
FIGURE 21-13. A lightning rod
allows charges from the clouds to be
grounded, rather than conducted
through a barn.
Negatively charged cloud
~ ~
@J
Positive ions in
atmosphere
store electric charge. In honor of the city in which he worked, it was
called a Leyden jar. The Leyden jar was used by Benjamin Franklin to
store the charge from lightning and in many other experiments.
As described previously, as charge is added to an object, the potential
between that object and Earth increases. For a given shape and size of
the object, the ratio of charge to potential difference, q/V, is a constant.
The constant is called the capacitance, C, of the object. For a small
sphere far from the ground, even a small amount of added charge will
increase the potential difference. Thus, C is small. The larger the
sphere, the greater the charge that can be added for the same increase
in potential difference, thus the larger the capacitance. Van Musschenbroek found a way of producing a large capacitance in a small device.
A device that is designed to have a specific capacitance is called a
capacitor. All capacitors are made up of two conductors, separated by
an insulator. The two conductors have equal and opposite charges. Capacitors are used in electrical circuits to store electric energy.
Capacitors often are made of parallel conducting sheets, or plates,
separated by air or another insulator. Commercial capacitors, Figure
21-14, contain strips of aluminum foil separated by thin plastic and are
tightly rolled up to save room.
The capacitance is independent of the charge on it. Capacitance can
be measured by placing charge q on one plate and - q on the other,
and measuring the potential difference, V, that results. The capacitance
is then found by using the equation
C
Capacitance is the ratio of charge
stored to potential difference.
Capacitors store energy in the electric
field between charged conductors.
.:«.
V
Capacitance is measured in farads, F, named after Michael Faraday.
One farad is one coulomb per volt (CN). Just as one coulomb is a large
amount of charge, one farad is an enormous capacitance. Capacitors
are usually between 10 picofarads (10 x 10-12 F) and 500 microfarads
(500 x 10-6 F).
FIGURE 21-14.
capacitors.
Various types of
21.2 Applications of the Electric Field
441
F. Y. I.
Michael Faraday, a self-educated man, was hired by chemist
Sir Humphrey Davy as a bottle
washer. As time went by, Faraday became an even greater scientist than Davy. Besides his
brilliant discoveries in the fields
of chemistry and physics, he was
also a popular lecturer. A lecture
for young people, The Chemical
History of a Candle, was published in book form and became
a classic.
Example Problem
Finding the Capacitance
from Charge and Potential Difference
A sphere has a potential difference between it and Earth of 60.0 V
when charged with 3.0 x 10-6 C. What is its capacitance?
Given: potential difference,
V = 60.0 VB'
q = 3.0 X 10-6 C
Solution: C
= 9.. = 3.0
V
= 5.0
X
Unknown:
capacitance, C
.
C
q
asic equation:
=
V
6
10- C = 5.0 X 10-8 C/V
60.0 V
10-8 F = 0.050 fLF
X
Practice Problems
13. A 27-fLF capacitor has a potential difference of 25 V across it. What
is the charge on the capacitor?
14. Both a 3.3-fLF and a 6.8-fLF capacitor are connected across a 15-V
potential difference. Which capacitor has a greater charge? What is
it?
15. The same two capacitors are each charged to 2.5 x 10-4 C. Across
which is the potential difference larger? What is it?
~ 16. A 2.2-fLF capacitor is first charged so that the potential difference is
6.0 V. How much additional charge is needed to increase the potential difference to 15.0 V?
........................
CONCEPT REVIEW
2.1 If an oil drop with too few electrons is motionless in a Millikan oil
drop apparatus,
a. what is the direction of the electric field?
b. which plate, upper or lower, is positively charged?
2.2 If the charge on a capacitor is changed, what is the effect on
a. the capacitance, C?
b. the potential difference, V?
2.3 If a large, charged sphere is touched by a smaller, uncharged
sphere, as in Figure 21-9, what can be said about
a. the potentials of the two spheres?
b. the charges on the two spheres?
2.4 Critical Thinking: Suppose we have a large, hollow sphere that has
been charged. Through a small hole in the sphere, we insert a
small, uncharged sphere into the hollow interior. The two touch.
What is the charge on the small sphere?
442
Electric Fields
CHAPTER
21 REVIEW·
.
------------~~--~~~~~=-~
SUMMARY
21.1 Creating
REVIEWING CONCEPTS
and Measuring
Electric
Fields
· An electric field exists around any charged object. The field produces forces on other charged
bodies.
· The electric field intensity is the force per unit
charge. The direction of the electric field is the
direction of the force on a tiny, positive test
charge.
· Electric field lines provide a picture of the electric field. They are directed away from positive
charges and toward negative charges.
21.2 Applications
of the Electric
Field
· Electric potential difference is the change in potential energy per unit charge in an electric field.
Potential differences are measured in volts.
· The electric field between two parallel plates is
uniform between the plates except near the
edges.
· Robert Millikan's experiments showed that electric charge is quantized and that the charge carried by an electron is - 1.6 x 10-19 C.
· Charges will move in conductors until the electric potential is the same everywhere on the conductor.
· A charged object can have its excess charge removed by touching it to Earth or to an object
touching Earth. This is called grounding.
· Electric fields are strongest near sharply-pointed
conductors.
· Capacitance is the ratio of the charge on a body
to its potential. The capacitance of a body is independent of the charge on the body and the
potential difference across it.
KEY TERMS
electric field
electric field lines
electric potential difference
volt
potential difference
grounding
capacitance
capacitor
1. Draw the electric field lines between
a. two like charges.
b. two unlike charges.
2. What are the two properties a test charge
must have?
3. How is the direction of an electric field defined?
4. What are electric field lines?
5. How is the strength of an electric field indicated with electric field lines?
6. What SI unit is used to measure electrical potential energy? electric potential?
7. What will happen to the electric potential energy of a charged particle in an electric field
when the particle is released and free to
move?
8. Define the volt in terms of the change in potential energy of a charge moving in an electric field.
9. Draw the electric field lines between two parallel plates of opposite charge.
10. Why does a charged object lose its charge
when it is touched to the ground?
11. A charged rubber rod placed on a table maintains its charge for some time. Why is the
charged rod not grounded immediately?
12. A metal box is charged. Compare the concentration of charge at the corners of the box to
the charge concentration on the sides.
13. In your own words, describe a capacitor.
APPLYING CONCEPTS
1. What happens to the size of the electric field
if the charge on the test charge is halved?
2. Does it require more energy or less energy to
move a fixed positive charge through an increased electric field?
3. Figure 21-15 shows three spheres with
charges of equal magnitude, but with signs as
shown. Spheres y and z are held in place but
sphere x is free to move. Initially sphere x is
equidistant from spheres y and z. Choose the
Chapter 21 Review
443
path that sphere x will follow, assuming
other forces are acting.
no
>tc
A
B
FIGURE 21-15. Use with Applying
Concepts 3.
4. What is the unit of potential difference in
terms of m, kg, s, and C?
5. What do the electric field lines look like when
the electric field has the same strength at all
points in a region?
6. When doing a Millikan oil drop experiment, it
is best to work with drops that have small
charges. Therefore, when the electric field is
turned on, should you try to find drops that are
moving fast or slow? Explain.
7. If two oil drops can be held motionless in a
Millikan oil drop experiment,
a. can yoy be sure that the charges are the
same?
b. the ratios of what two properties of the
drops would have to be equal?
8. l)m and Sue are holding hands when they are
given a charge while they are standing on an
insulating platform. Tim is larger than Sue.
Who has the larger amount of charge or do
they both have the same amount?
9. Which has a larger capacitance, a t-ern diameter or a to-ern diameter aluminum
sphere?
10. How can you store a different amount of
charge in a capacitor?
PROBLEMS
21.1 Creating
and Measuring
Electric
Fields
The charge on an electron is -1.60 x 10-19 C.
1. A positive charge of 1.0 x 10 - 5 C experiences a force of 0.20 N when located at a certain point. What is the electric field intensity at
that point?
444
Electric Fields
I
2. What charge exists on a test charge that experiences a force of 1.4 x 10-8 N at a point
where the electric field intensity is 2.0 x 10-4
N/C?
3. A test charge has a force of 0.20 N on it when
it is placed in an electric field intensity of
4.5 x 105 N/C. What is the magnitude of the
charge?
4. The electric field in the atmosphere is about
150 N/C, downward.
a. What is the direction of the force on a positively-charged particle?
b. Find the electric force on a proton with
charge +1.6 x 10-19 C.
c. Compare the force in b with the force of
gravity on the same proton (mass 1.7 x
10-27 kg).
~ 5. Electrons are accelerated by the electric field
in a television picture tube, Table 21-1.
a. Find the force on an electron.
b. If the field is constant, find the acceleration
of the electron (mass = 9.11 x 10-31 kg).
~ 6. A lead nucleus has the charge of 82 protons.
a. What is the direction and magnitude of the
electric field at 1.0 x 10-10 m from the nucleus?
b. Use Coulomb's law to find the direction
and magnitude of the force exerted on an
electron located at this distance.
7. Carefully sketch
a. the electric field produced by a + 1.0 I-LC
charge.
b. the electric field due to a + 2.0 I-LCcharge.
Make the number of field lines proportional
to the change in charge.
8. Charges X, Y, and Z are all equidistant from
each other. X has a + 1.0 I-LC charge, Y a
+ 2.0 I-LC charge, and Z a small negative
charge.
a. Draw an arrow showing the force on
charge Z.
b. Charge Z now has a small positive charge
on it. Draw an arrow showing the force on
it.
9. A positive test charge of 8.0 x 10-5 C is
placed in an electric field of 50.0-N/C intensity. What is the strength of the force exerted
on the test charge?
21.2 Applications
of the Electric
Field
10. If 120 J of work are done to move one
coulomb of charge from a positive plate to a
negative plate, what voltage difference exists
between the plates?
11. How much work is done to transfer 0.15 C of
charge through a potential difference of
9.0 V?
12. An electron is moved through a potential difference of 500 V. How much work is done on
the electron?
13. A 12-V battery does 1200 J of work transferring charge. How much charge is transferred?
~ 14. A force 0.053 N is required to move a charge
of 37 f.LC a distance of 25 cm in an electric
field. What is the size of the potential difference between the two points?
15. The electric field intensity between two
charged plates is 1.5 x 103 N/C. The plates
are 0.080 m apart. What is the potential difference, in volts, between the plates?
16. A voltmeter indicates that the difference in potential between two plates is 50.0 V. The
plates are 0.020 m apart. What electric field
intensity exists between them?
17. A negatively-charged oil drop weighs 8.5 x
10-15 N. The drop is suspended in an electric
field intensity of 5.3 x 103 N/C.
a. What is the charge on the drop?
b. How many electrons does it carry?
~ 18. In an early set of experiments (1911), Millikan
observed
that
the following
measured
charges, among others, appeared at different
times on a single oil drop. What value of elementary charge can be deduced from these
data?
f. 18.08 X 10-19 C
a. 6.563 x 1O-19
19 C
10g. 19.71 X 10-19 C
b. 8.204 x
19 C
10h. 22.89 X 10-19 C
c. 11.50 x
19 C
i. 26.13 X 10.-19 C
10d. 13.13 x
e. 16.48 x 1O-19C
19. A capacitor that is connected to a 45.0-V
source contains 90.0 f.LC of charge. What is
the capacitor's capacrtance?
20. A 5.4-f.LF capacitor is charged with 2.7 x
10-3 C. What potential difference exists
across it?
21. What is the charge in a 15.0-pf capacitor
when it is connected across a 75.0-V source?
~ 22. The energy stored in a capacitor with capacitance C, having a potential difference V, is
given by W = V2CV2. One application is in
the electronic photoflash or strobe light. In
such a unit, a capacitor of 10.0 f.LFis charged
to 3.00 x 102 V. Find the energy stored.
c
~ 23. Suppose it took 30 s to charge the capacitor
in the previous problem.
a. Find the power required to charge it in this
time.
b. When this capacitor is discharged through
the strobe lamp, it transfers all its energy
in 1.0 x 10-4 s. Find the power delivered
to the lamp.
c. How is such a large amount of power possible?
~ 24. Lasers are used to try to produce controlled
fusion reactions that might supply large
amounts of electrical energy. The lasers require brief pulses of energy that are stored in
large rooms filled with capacitors. One such
room has a capacitance of 61 x 10-3 F
charged to a potential difference of 10 kV.
a. Find the energy stored in the capacitors,
given W = V2CV2.
b. The capacitors are discharged in 10 ns
'(1.0 x 10-8 s). What power is produced?
c. If the capacitors are charged by a generator with a power capacity of 1.0 kW, how
many seconds will be required to charge
the capacitors?
USING LAB SKILLS
1. In the Pocket Lab on page 426, does the angle
of the thread show the direction of the electric
field? Explain. Make a sketch as part of your
answer.
2. In the Pocket Lab on page 436, you measured
the mass of three groups of balls. Based on
these measurements, predict what other values should occur if you continue to measure a
random number of balls.
THINKING
PHYSIC-LY
Why are many parts of stereo components covered with metal boxes or containers?
Chapter 21 Review
445