Download Chapter 4: The Fundamental Interactions

4. The Fundamental Interactions
So far in our study of motion, we have learned that
forces occur in interactions and cause accelerations. By
itself, this information is of limited use until we learn
how to identify interactions and predict the important
features of the resulting forces.
All the forces we directly encounter in ordinary life
result from only two kinds of interactions—gravitational and electromagnetic. The other fundamental
interactions are nuclear, which are important only
when the interacting objects are as close together as
inside the nucleus of an atom. These will be discussed
in a later chapter.
As we discuss the electromagnetic interaction, we
will also begin to examine the question of what matter is
made. We will find that all matter, living as well as nonliving, is composed of electrically charged particles.
The electromagnetic interaction between these is responsible for most of the forces we encounter. We begin our
discussion of interactions with a consideration of some
of the manifestations of the gravitational interaction.
Falling Objects
Consider the motion of an object, such as a baseball, dropped from a great height. As it falls, its speed
steadily increases. If we were to make careful measurements, we would find that, in the absence of air friction, its speed would increase at a constant rate. After 1
sec, its speed would be 35 kilometers/hour; after 2 seconds, 70 kilometers/hour; and after 3 seconds, 105 kilometers/hour. The speed would increase at 35 kilometers/hour every second as long as it falls. This rate of
acceleration is sometimes designated by the symbol g.
The falling ball is clearly accelerating. If its motion
is in harmony with the Second Law of Motion, and it is,
some downward force must be acting upon it. The force
that causes this acceleration is called the weight of the
ball (Fig. 4.1).
The next step is to drop a different object (e.g., a
large anchor) from the same height. Before actually
doing the experiment, we might expect the anchor to
drop more rapidly than the baseball, since it obviously
has more weight. But nature does not always behave
the way we expect. In this case, the acceleration of the
anchor is exactly the same as that of the baseball!
Figure 4.1. Why does a falling ball accelerate?
Somehow the force causing the acceleration, the weight
of the anchor, has increased in exactly the same ratio as
its mass so that the resulting acceleration, determined
by force divided by mass, does not change (Fig. 4.2).
This surprising result is true for all objects near the
surface of the earth. Even light objects such as feathers
and sheets of paper have exactly the same acceleration
when allowed to fall in the absence of air resistance.
The free-fall acceleration, g, is the same for all objects.
This must mean that weight (the force causing the acceleration) and mass are proportional. If the mass of one
object is two times the mass of another, its weight is
larger in exactly the same ratio. The acceleration (force
divided by mass) is then the same for both. These
results suggest the following conclusion:
Every object near the surface of the earth experiences a downward force, called its weight, the
strength of which is exactly proportional to its mass.
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Clear evidence also shows that the moon pulls on
the earth. The most apparent results of this force are the
lunar tides in which the level surfaces of oceans rise and
fall as the moon passes overhead. The earth’s attraction
to the moon also causes the earth to accelerate slightly.
Such accelerations are measured routinely by sophisticated navigational instruments such as those used on
submarines.
An additional feature of the moon’s acceleration
worth noting is calculated by using a mathematical definition of acceleration and measurements of the moon’s
orbit. The moon’s sideways acceleration is almost
exactly 1/3,600 the acceleration of an object falling near
the earth’s surface, so the earth’s pull on the moon must
be only 1/3,600 as strong as it would be if the moon
were moved to the earth’s surface.
The Law of Universal Gravitation
Isaac Newton was the first to suggest that the
attraction of the moon to the earth is due to the same
kind of interaction that causes free objects near the earth
to fall. These are called gravitational interactions.
The observations we have described tell us much
about the forces resulting from gravitational interaction.
First, the two interacting objects always attract each
other. Each force is proportional to the mass of the
object on which it acts because g is the same for all
objects. The two forces in each interaction obey the
Third Law of Motion. (Remember the lunar tides and
the acceleration of the earth due to the moon’s attraction.) Finally, the force is weaker if the two objects are
farther apart, since the moon’s acceleration is only
1/3,600 as much as it would be if it were near the earth’s
surface. In fact, the strength of the force depends on the
square of the distance between the centers of the interacting objects. The moon is 60 times farther from the
center of the earth than is the earth’s surface. Notice
that 602 is 3600, the observed factor by which the gravitational force on the moon is diminished.
With these insights, Newton suggested that every
Figure 4.2. Why do a falling anchor and a falling ball
accelerate at the same rate when air resistance is not
important?
From these observations alone, we do not really
know where this force comes from, but apparently
every object is pulled toward the earth. We might suspect that some kind of interaction between the object
and the earth is responsible.
The Moon’s Orbit
The moon circles the earth in an almost perfect circle every 27.3 days. Since it is not moving in a straight
line, we know that it is accelerating and that its acceleration is caused by some force. What can we say about
the force?
Your understanding of the last chapter lets you
know immediately that the moon is experiencing a force
sideways to its direction of motion that causes the continuous change in the direction of its motion. Further,
you know that the force is directed toward the center of
the circular path. The moon moves as if it, too, is being
pulled toward the earth (Fig. 4.3).
m
M
Figure 4.4. Every object is attracted to every other
object through gravitational interaction. The two forces
have the same strength.
Figure 4.3. Something must be pulling or pushing the
moon toward the earth. How do we know? What is it?
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object in the universe interacts with every other object
through gravitational interaction (Fig. 4.4). Since the
strengths of the resulting forces depend on mass, they
are ordinarily too small to be noticed for most objects.
Only if one of the interacting objects has a large mass,
like the earth, does the force become appreciable.
This Universal Law of Gravitation (or the Law
of Gravity) can be summarized as follows:
Every object in the universe attracts every
other object by a long-range gravitational
interaction that obeys Newton’s Third Law.
The strength of the attractive force, F, varies
with the masses, M and m, of the two objects
and the distance, d, between their centers
according to the relationship
F"
GmM .
d2
Newton’s hypothesis is subject to experimental verification. It was confirmed in every detail over 70 years
after Newton’s death by Henry Cavendish (1731-1810),
who finally developed a method of measuring the gravitational attraction between such ordinary-sized objects
as two large lead balls. Earlier support had come by
studying the planets and their moons, whose motions
through space can be explained in terms of Newton’s
Universal Law of Gravitation.
The number G that appears in the equation for the
strength of the gravitational force is called the gravitational constant. It must be measured experimentally
and is so small that the mutual attractive gravitational
force between two 100-kilogram balls placed 30 centimeters apart would be equivalent to the earth weight of
only 0.01 gram of mass. (The actual value of G is 6.67
! 10–11 in the metric system of units.) No wonder we
ordinarily do not notice these forces, which were measured only in fairly recent times.
Figure 4.5. Two rubbed rubber rods or two rubbed glass
rods repel each other. Yet a rubber rod and a glass rod
are attracted. Why?
increases as the rods get closer.
A new feature is revealed, however, when we bring
a charged glass rod near a charged rubber rod. The two
dissimilar rods attract each other with a force that
becomes larger as the rods come closer together. We are
dealing with something more complicated than gravity,
since these forces can be either attractive or repulsive,
depending on the circumstances. Other kinds of materials can be electrified by rubbing. When they are, pairs
of similar rods always repel each other. Some, however, are attracted to a charged rubber rod and some are
repelled by it. Those that are attracted to the charged
rubber rod are repelled by a charged glass rod and vice
versa. Those attracted to the rubber rod are said to be
positively charged; those attracted to the glass, negatively charged (Fig. 4.5).
The explanation of these experiments requires two
new broad insights. First, we need to know more about
how materials are made and what it is that changes
when they become charged. The second major part of
the puzzle has to do with the law governing the interac-
Some Simple Experiments with Electricity
It has been known, at least since early Greek times,
that certain pairs of materials become “electrified” or
“charged” when they are rubbed together. Suppose we
rub one end of a hard rubber rod with a piece of fur and
then hang the rod from a string without touching the
rubbed end. Then we similarly rub one end of a second
rod and hold it near the first. You will see from the
motion of the hanging rod that the rods repel each other
even when they are some distance apart. There is an
interaction between the two rods. Further careful testing would show that the repulsion becomes greater as
the two rods come closer together.
Two glass rods that are rubbed with silk react similarly. The glass rods repel each other with a force that
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tion itself. What determines the strength and direction
of the resulting forces?
force, and speed, can have any value and are said to be
continuous.
In many materials some of the electrons can be
removed from the surface by rubbing. When rubber is
rubbed with fur, some of the electrons in the fur are
transferred to the rubber, which becomes negatively
charged. (The fur and rod are attracted to each other,
incidentally.) The electrons carry so little mass that the
objects seem the same as before, except that they are
now electrically charged. Protons, because of their larger mass, are held rigidly in place in all solid materials.
This picture of matter might be termed the
Electrical Model of Matter. It leaves many questions
unanswered (e.g., how these charged particles are
arranged in matter, how they combine to create the
almost numberless kinds of materials found in living
and nonliving matter, and what happens when materials
change form). But it does provide an adequate model
for explaining a wide range of experiences. We can
summarize the model as follows:
The Electrical Model of Matter
An important conclusion of these experiments is
that matter is made of more basic pieces. Rubber, glass,
silk, fur, and all other materials presumably have some
important constituents in common. The experiments
suggest at least two kinds.
Many objects do not seem to be either attracted or
repelled by charged objects, yet can be charged by rubbing. This suggests that the materials normally contain
both kinds of constituents. When the constituents occur
in equal amounts, they cancel each other’s effects; the
material is not charged, and is said to be electrically
neutral. When this balance is disturbed by rubbing one
of the constituents either off or onto the object, for
example, the object becomes charged. If it has more of
one constituent, it is said to have a positive charge; if
more of the other, a negative charge (Fig. 4.6).
All matter contains two kinds of electrically
charged particles: positive protons and negative
electrons. Electrons have little mass and can be
quite mobile and transferable from one object to
another. Protons are held rigidly in place in
solid materials. Objects that have equal numbers of protons and electrons are electrically
neutral. Objects with more electrons than protons are negatively charged. Those with fewer
electrons than protons are positively charged.
The amount of extra charge of either kind is
called the “charge of an object.”
a
b
c
The Electrical Force Law
Figure 4.6. All matter contains electric charge. The
object in (a) is electrically neutral, in (b) it is positively
charged, and in (c) it is negatively charged.
By now you have probably guessed the main features of the electrical interaction. Objects with the same
kind of charge repel each other. (Remember that in our
experiments identical rods always repelled each other.)
Objects with opposite charges—one positive, the other
negative—attract each other. The forces, attractive or
repulsive, become stronger when the charged objects
are closer together. Careful measurements have shown
that the strength of the force varies with separation in
exactly the same way as for gravitational force—
inversely as the square of the distance between the interacting objects. The strength also depends on the
amount of extra charge possessed by each object,
increasing in exact proportion to the net charge on each.
Finally, electrical forces obey Newton’s Third Law.
These important features of the electrical interaction are
summarized in the following statement (Fig. 4.7):
More sophisticated research that is discussed later
reveals that protons are an important constituent of all
matter. These are tiny, dense particles in the center of all
atoms. All protons are exactly alike, and each carries
one unit of positive electric charge. The negative charge
in matter is supplied by electrons, each of which can balance the positive charge of a proton exactly. Electrons
have little mass—only about 1/1,836 that of protons.
The unit of charge used in calculations is the
coulomb, equivalent to the charge of about 6.0 ! 1018
protons. We could measure electric charge by simply
counting the number of excess electrons or protons, but
this is usually impractical because of the large numbers
involved.
Electric charge has one property that we have not
encountered previously. It is discrete; that is, it occurs
only in multiples of a fundamental unit, the charge of a
single proton. Other physical quantities, such as mass,
Pairs of objects with similar charges repel each
other and pairs with dissimilar charges attract
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each other with forces, F, that obey Newton’s
Third Law and whose strength depends on the
net charges, q and Q, on the objects and the
distance, d, between them according to the
relationship.
kqQ
.
F"
d2
Imagine water flowing through a pipe loosely filled
with gravel. Electric current in a metal wire is similar.
The moving water represents the electrons; the stationary gravel represents the positive charges in the wire
(together with the rest of the electrons, which are not
free to move about). Notice that no part of the wire is
charged, because there are always equal numbers of
positive and negative charges in any part of the metal.
Electric current flows in a circuit in which a battery
plays the role that a pump plays in our water and pipe
analogy (see Fig. 4.8). The circuit must be completed by
closing the switch. Batteries produce a direct current of
electrons that flows in only one direction through the circuit. The wall socket into which household appliances are
connected is like the battery, except that it reverses the
direction of current flow 60 times per second. Such a current flow is called alternating current.
d
Q
Q
q
q
Light
Q
q
Figure 4.7. Every charged object is attracted or repelled by
every other charged object through the electrical interaction. The two forces have the same strength in every case.
(Only the excess charges are shown in these diagrams.)
Switch
+
The constant, k, that appears in the strength equation is called the electrical force constant. As with the
gravitational constant, it must be measured experimentally. The electrical force constant is large; its numerical value is about 9 ! 109 in the metric system. This
means that the electrical force is easy to demonstrate,
whereas the gravitational force between ordinary
objects can be observed only in sensitive and careful
experiments. In fact, the experiments described earlier
involve the transfer of only a small fraction (about 1 out
of every 1012) of the electrons actually present. If separating all the electrons from the protons in a single copper penny were possible, and the electrons and protons
were placed 100 meters (about the length of a football
field) from each other, the collection of particles would
attract each other with a force of about 1012 tons. The
electrical force can be strong indeed.
Conducting
wire
Battery
Figure 4.8. Electric current flows in a complete circuit.
The arrows show the direction of motion of the electrons.
What are the purposes of the switch and the battery?
Electromagnetic Forces
The electrical interaction described to this point is
accurate for charges that are at rest. The total interaction
between charged particles depends on the motion of the
particles, as well as the factors already discussed. The
changes that occur when charges are moving result in magnetic forces, some of which you have undoubtedly encountered. They are usually not important if the electrical interaction is operating, and they result only in small
motion-dependent corrections. They can become important, however, when charges are moving inside electrically
neutral objects, such as when current flows through a wire.
The complete interaction due to electric charge is
Electric Currents
Some of the electrons are free to move on the surface or through the interior of some materials which are
known as conductors. Insulators are materials that do
not permit this free interior motion of electrons.
Semiconductors contain a few free electrons, but not as
many as conductors.
Moving charged particles form an electric current.
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known as the electromagnetic interaction. It includes
the electrical interaction between charged particles,
either moving or at rest, as well as the magnetic interactions between moving charged particles.
forces exerted by gasoline and steam engines.
Now that we have studied the fundamental motion
and force laws one at a time, we are ready to consider
some real applications.
Summary
STUDY GUIDE
Chapter 4: The Fundamental Interactions
Four fundamental interactions—gravitational, electromagnetic, weak, and strong (or nuclear)—cause all
the forces we know about. The gravitational interaction, together with the laws of motion, explains the
motion of falling objects and is the source of the force
called weight. The electromagnetic interaction is associated with all the other forces governing the motion of
objects larger than atomic nuclei.
The five important laws are the three laws of
motion and the two macroscopic force laws. Together
they make a tidy package that describes and predicts
with amazing accuracy the motions of objects ranging
in size from atoms to clusters of galaxies. All five are
needed before we are ready to apply any of them.
The gravitational acceleration of an object is the
acceleration it would experience if the gravitational
force were the only force acting on it. Gravitational
acceleration depends on the location of an object, but
not on its mass. That is, all objects have the same gravitational acceleration at a given point in space.
The weight of an object is the gravitational force
acting on it. It depends on the object’s location. The
weight of an object near the surface of the moon would
be about 1/6 its weight near the earth. Weight on Jupiter
is 2.7 times earth weight and weight near the sun’s surface is 28 times earth weight.
It is possible to measure the very small gravitational attraction between ordinary-sized objects by using a
Cavendish balance. Such measurements provide direct
experimental evidence supporting the Universal Law of
Gravitation.
The electric force can be a very strong force, even
between ordinary objects. If it were possible to separate
all the negative and positive charges in a penny from
each other, for example, they would attract each other
with a force of more than one trillion tons at a distance
of 100 meters.
Several common demonstrations illustrate the
Electric Force Law and the motion of electric charges:
walking across a rug, then touching a metal doorknob;
lightning; the attraction and repulsion of rubber and
glass rods rubbed with fur and silk; the operation of an
electroscope when touched by a charged object.
The electrical force is the “glue” that holds the particles of matter together. It is responsible for all the
contact forces we ordinarily experience. Examples are
friction; atmospheric pressure; the strength of bridges
and buildings; the impact forces that occur, for example,
when billiard balls or automobiles collide; and the
A. FUNDAMENTAL PRINCIPLES
1. The Universal Law of Gravitation: Every object
in the universe attracts every other object by a longrange gravitational interaction that obeys Newton’s
Third Law. The strength of the attractive force, F,
varies with the masses, M and m, of the two objects
and the distance, d, between their centers according
to the relationship
F"
2.
GmM .
d2
The Electric Force Law: Pairs of objects with
similar charges repel each other and pairs with dissimilar charges attract each other with forces that
obey Newton’s Third Law and whose strength
depends on the net charges, q and Q, on the objects
and the distance, d, between them according to the
relationship
F"
kqQ .
d2
B. MODELS, IDEAS, QUESTIONS, OR
APPLICATIONS
1.
2.
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The Newtonian Model
(sometimes, the
Newtonian Synthesis): The model based on
Newton’s three laws of motion and the Universal
Law of Gravitation which explains the motions of
the heavens as well as the terrestrial motions of
common experience. The Newtonian Model when
applied to the motions of the planets replaces the
medieval model which placed the earth at the center of the solar system and the universe.
Electrical Model of Matter: All matter contains
two kinds of electrically charged particles: positive
protons and negative electrons. Electrons have little mass and can be quite mobile and transferable
from one object to another. Protons are held rigidly in place in solid materials. Objects that have
equal numbers of protons and electrons are electrically neutral. Objects with more electrons than
protons are negatively charged. Those with fewer
electrons than protons are positively charged. The
amount of extra charge of either kind is called the
“charge of the object.”
3.
4.
5.
6.
7.
8.
C
1.
2.
3.
4.
5.
6.
Why do both heavy and light things accelerate at
the same rate when only the gravitational force is
acting on them?
How can the acceleration of the moon and the
acceleration of a falling apple be accounted for by
the same Universal Law of Gravitation?
What determines the strength of all gravitational
forces?
What is the Electrical Model of Matter?
What determines the strength of electrical forces?
What interactions are responsible for all of the
forces we observe in ordinary life experiences?
8.
9.
F"
GmM .
d2
10. Insulator (specifically, of electricity): A substance
which does not readily allow an electric current to
flow through it. The opposite of a conductor. Glass
is an insulator.
11. Semiconductors: Materials whose electrical conducting properties place them somewhere midway
between conductors and insulators. Silicon is a
semiconductor.
12. Weight: The gravitational force of attraction of a
very massive object, usually a planet or moon, for
a less massive object on or near its surface.
GLOSSARY
Circuit: A connected, continuous path along
which electrical charge flows to produce an electrical current.
Conductor (specifically, of electricity): A substance which readily allows an electric current to
flow through it. The opposite of an insulator (nonconductor). Copper wire is a conductor.
Continuous: Varying smoothly without distinct
parts or discontinuous elements. Used here to
mean the opposite of “discrete.”
Coulomb: The unit of charge used in calculations,
D. FOCUS QUESTIONS
1. In each of the following situations:
a. Describe what would be observed.
b. Name and state in your own words the fundamental principle(s) that could explain what would
happen.
c. Explain what would happen in terms of the fundamental principle(s).
(1) A penny and a feather are caused to fall
toward the earth in a vacuum tube. They start
to fall at the same time.
(2) Suppose an elephant and a feather were to
fall from a high cliff at exactly the same time.
If air friction could be ignored, what would
happen?
(3) A rubber rod is rubbed with fur and placed
on a wire rack suspended by a string. A second
rubber rod that has been rubbed with fur is
brought nearby. The second rod is then taken
away and a glass rod that has been rubbed with
a vinyl sheet is brought nearby. (Note: the rubber rod acquires extra electrons. The glass rod
loses electrons.)
equivalent to the charge of about 6 ! 1018 protons.
Discrete: Separate or individually distinct, consisting of distinct parts or discontinuous elements.
Used here to mean the opposite of “continuous” or
smoothly varying. The electric charge of an electron is described as discrete since it cannot be
smoothly subdivided into smaller parts.
Electrical Force Constant: The electrical force
constant is usually represented by the symbol k. It
is a constant of proportionality in the Electric Force
Law which connects the strength of the electrical
force to its dependence on the charges of objects
and their separations.
F"
7.
stant is usually represented by the symbol G. It is
a constant of proportionality in Newton’s Universal
Law of Gravitation which connects the strength of
the gravitational force to its dependence on the
masses of the objects and their separations.
kqQ .
d2
Electric Current: A coherent motion of electrical
charges constitutes an electrical current. The
motion of electrons along or through a copper wire
is an example of an electrical current. If the flow is
only in one direction, the current is said to be
direct. If the current periodically reverses its direction of flow, the current is said to be alternating.
Free-fall Acceleration, g: The acceleration of a
falling object on which the only significant force is
the gravitational force. Near the surface of the
earth, the free-fall acceleration is about 35 kilometers per hour per second.
Gravitational Constant: The gravitational con-
E. EXERCISES
4.1. The earth pulls on you with a gravitational
force of attraction, your weight. Describe the “reaction”
to this force. Show that your answer is consistent with
the Third Law of Motion.
4.2. If you are pulling on the earth with a gravitational force, why doesn’t the earth move in the same
way you do in response to that force? Show that your
answer is consistent with the Second and Third Laws of
Motion.
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4.3. Why does an object weigh less near the surface of the moon than near the surface of the earth?
4.14. Describe the important properties of a proton.
4.15.
electron.
4.4. The sun has much more mass than the earth
(about 330,000 times as much). Why aren’t we pulled
toward the sun with 330,000 times as much force as we
are toward the earth?
Describe the important properties of an
4.16. Describe the Electrical Model of Matter.
4.17. What is meant when we say that electric
charge is discrete?
4.5. Compare the weights of an object in three
locations:
(a) near the surface of the earth,
(b) near the surface of the moon, and
(c) in a place outside the solar system where there
are almost no gravitational forces.
4.18. Describe how the electrons and protons in an
atom could be held together by the electrical force.
4.19. Explain why you experience a repulsive
force when you slap a table with your hand.
4.6. How does the mass of the object in the previous
exercise change as it is taken to the same three locations?
4.20. Precisely what is electric current?
4.7. Compare the definitions of weight and mass.
Can you see why the weight of an object can change
from place to place while its mass does not? Explain
how this can be so.
4.21. A rubber rod rubbed with fur and then
brought near a second, similarly prepared, rubber rod
can illustrate the Electric Force Law.
(a) Describe what happens when the two rods are
brought near each other.
(b) Explain how the observed results illustrate the
Electric Force Law.
(c) What happened to the rubber rods when they
were rubbed with fur?
(d) What would happen if the rubber rods were
brought near a glass rod which had been rubbed
with silk?
(e) What additional feature of the Electric Force
Law is illustrated by this second experiment?
(f) What happened to the glass rod when it was
rubbed with silk?
4.8. A cannonball originally at rest and a marble
originally at rest are dropped in a vacuum from the same
height at the same time.
(a) What happens when they are dropped?
Compare the speed and acceleration of the cannonball with that of the marble.
(b) Is the gravitational force of attraction larger on
the cannonball than it is on the marble? Justify
your answer using a fundamental law.
(c) Does the cannonball require a larger force to
provide the same acceleration as the marble?
Justify your answer using the Second Law of
Motion.
(d) Show that your answers to (a), (b), and (c) are
consistent with each other.
4.22. State the Electric Force Law and explain its
meaning.
4.23. How do we know that the Electrical Model of
Matter and the Electric Force Law are valid descriptions
of nature?
4.9. State the Universal Law of Gravitation and
explain its meaning in your own words.
4.10. A small ball is dropped from the edge of a cliff.
One-tenth of a second later a much heavier ball is
dropped from the same position. Ignoring the effects of
air friction, can the second ball overtake the first? Justify
your answer using fundamental laws or principles.
4.24. When a glass rod is rubbed with rubber, it
becomes positively charged. This is because
(a) protons are transferred from rubber to glass.
(b) protons are transferred from glass to rubber.
(c) electrons are transferred from glass to rubber.
(d) electrons are transferred from rubber to glass.
(e) electrons and protons annihilate each other.
4.11. Describe an experiment that demonstrates
that there are two kinds of electric charge.
4.12. What is meant when we say that an object is
“charged”?
4.13. Describe what happens when a glass rod
becomes positively charged by being rubbed with silk.
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