copyright 2006 Benjamin Crowell

rev. May 22, 2019

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2

Brief Contents

1 Conservation of Mass and Energy

7

2 Conservation of Momentum

37

3 Conservation of Angular Momentum

61

4 Relativity

71

5 Electricity

93

6 Fields

111

7 The Ray Model of Light

131

8 Waves

161

For a semester-length course, all seven chapters can be covered. For a shorter course, the book is designed so that chapters 1, 2, and 5 are the only ones that are required for continuity; any of the others can be included or omitted at the instructor’s discretion, with the only constraint being that chapter 6 requires chapter 4.

3

Contents

Momentum compared to kinetic energy,

45.—Force, 46.—Motion in two dimen-

sions, 49.

2.4 Newton’s triumph . . . . . . . .

53

2.5 Work . . . . . . . . . . . . .

57

Problems . . . . . . . . . . . . .

59

3 Conservation of Angular Mo-

mentum

3.1 Angular momentum . . . . . . .

61

1 Conservation of Mass and

3.2 Torque

. . . . . . . . . . . .

64

Energy

Torque distinguished from force, 66.

3.3 Noether’s theorem for angular mo-

1.1 Symmetry and conservation laws .

7

mentum . . . . . . . . . . . . . .

67

1.2 Conservation of mass . . . . . .

9

Problems . . . . . . . . . . . . .

68

1.3 Review of the metric system and con-

versions . . . . . . . . . . . . . .

11

The metric system, 11.—Scientific nota-

tion, 12.—Conversions, 13.

1.4 Conservation of energy

. . . . .

15

Energy, 15.—The principle of inertia, 16.—

Kinetic and gravitational energy, 20.—

Energy in general, 21.

1.5 Newton’s law of gravity . . . . . .

25

1.6 Noether’s theorem for energy . . .

27

1.7 Equivalence of mass and energy

.

29

Mass-energy,

29.—The

correspondence

principle, 31.

Problems . . . . . . . . . . . . .

33

4 Relativity

4.1 Relativity according to Einstein . .

71

Time is not absolute, 71.—Distortion of

time and space, 73.—The G factor, 77.

4.2 Speeds in relativity . . . . . . .

80

The universal speed c,

80.—Velocities

don’t simply add and subtract., 80.—A

universal speed limit, 80.—Light travels

2 Conservation of Momentum

at c., 80.—The Michelson-Morley experi-

2.1 Translation symmetry . . . . . .

38

ment, 81.

2.2 The principle of inertia . . . . . .

39

4.3 Dynamics . . . . . . . . . . .

85

Symmetry and inertia, 39.

Momentum, 85.—Equivalence of mass and

2.3 Momentum . . . . . . . . . . .

40

energy, 87.

Conservation

of

momentum,

40.—

Problems . . . . . . . . . . . . .

89

4

5 Electricity

Problems . . . . . . . . . . . . . 127

5.1 The quest for the atomic force . . .

94

5.2 Charge, electricity and magnetism .

95

Charge, 95.—Conservation of charge, 97.—

Electrical forces involving neutral objects,

97.—The atom, and subatomic particles,

98.—Electric current, 98.

5.3 Circuits . . . . . . . . . . . . 100

5.4 Voltage . . . . . . . . . . . . 101

The volt unit, 101.

7 The Ray Model of Light

7.1 Light rays . . . . . . . . . . . 131

5.5 Resistance . . . . . . . . . . . 104

The nature of light, 132.—Interaction of

Applications, 105.

light with matter, 135.—The ray model of

light, 136.—Geometry of specular reflec-

Problems . . . . . . . . . . . . . 109

tion, 138.

7.2 Applications . . . . . . . . . . 140

The inverse-square law, 140. —Parallax, 142.

7.3 ?The principle of least time for reflec-

tion . . . . . . . . . . . . . . . . 146

7.4 Images by reflection . . . . . . . 147

A virtual image, 147.—Curved mirrors,

148.—A real image, 149.—Images of im-

ages, 151.

Problems . . . . . . . . . . . . . 156

6 Fields

6.1 Farewell to the mechanical universe 111

Time delays in forces exerted at a dis-

tance, 112.—More evidence that fields of

force are real: they carry energy. , 113.—

The gravitational field, 113.—Sources and

sinks, 114.—The electric field, 115.

8 Waves

6.2 Electromagnetism . . . . . . . . 115

8.1 Vibrations . . . . . . . . . . . 161

8.2 Wave motion . . . . . . . . . . 164

Magnetic interactions, 115.—Relativity re-

1. Superposition, 164.—2. The medium

quires magnetism, 116.—Magnetic fields,

is not transported with the wave., 166.—3.

119.

A wave’s velocity depends on the medium. ,

167.—Wave patterns, 168.

6.3 Induction. . . . . . . . . . . . 122

8.3 Sound and light waves . . . . . . 168

Electromagnetic waves, 125.

Sound waves, 169.—Light waves, 170.

5

8.4 Periodic waves . . . . . . . . . 170

172.

Problems . . . . . . . . . . . . . 175

Period and frequency of a periodic wave,

170.—Graphs of waves as a function of po-

sition, 171.—Wavelength, 172.—Wave ve-

Photo Credits 177

locity related to frequency and wavelength,

Hints and Solutions 179

6

Chapter 1

Conservation of Mass and

Energy

1.1 Symmetry and conservation laws

Even before history began, people must already have noticed

certain facts about the sky. The sun and moon both rise in the east

and set in the west. Another fact that can be settled to a fair degree

of accuracy using the naked eye is that the apparent sizes of the sun

and moon don’t change noticeably. (There is an optical illusion that

makes the moon appear bigger when it’s near the horizon, but you

can easily verify that it’s nothing more than an illusion by checking

its angular size against some standard, such as your pinkie held

at arm’s length.) If the sun and moon were varying their distances

from us, they would appear to get bigger and smaller, and since they

don’t appear to change in size, it appears, at least approximately,

that they always stay at the same distance from us.

From observations like these, the ancients constructed a scientific

model, in which the sun and moon traveled around the earth in

perfect circles. Of course, we now know that the earth isn’t the

a / Due to the rotation of the

center of the universe, but that doesn’t mean the model wasn’t

earth,

everything

in

the

sky

useful. That’s the way science always works. Science never aims

appears to spin in circles. In this

to reveal the ultimate reality. Science only tries to make models of

time-exposure photograph, each

reality that have predictive power.

star appears as a streak.

Our modern approach to understanding physics revolves around

the concepts of symmetry and conservation laws, both of which are

demonstrated by this example.

The sun and moon were believed to move in circles, and a circle

is a very symmetric shape. If you rotate a circle about its center,

like a spinning wheel, it doesn’t change. Therefore, we say that the

circle is symmetric with respect to rotation about its center. The

ancients thought it was beautiful that the universe seemed to have

this type of symmetry built in, and they became very attached to

the idea.

A conservation law is a statement that some number stays the

same with the passage of time. In our example, the distance between

the sun and the earth is conserved, and so is the distance between

the moon and the earth. (The ancient Greeks were even able to

7

determine that earth-moon distance.)

b / Emmy Noether (1882-1935). The daughter of a prominent German

mathematician, she did not show any early precocity at mathematics —

as a teenager she was more interested in music and dancing. She re-

ceived her doctorate in 1907 and rapidly built a world-wide reputation, but

the University of G öttingen refused to let her teach, and her colleague

Hilbert had to advertise her courses in the university’s catalog under his

own name. A long controversy ensued, with her opponents asking what

the country’s soldiers would think when they returned home and were ex-

pected to learn at the feet of a woman. Allowing her on the faculty would

also mean letting her vote in the academic senate. Said Hilbert, “I do

not see that the sex of the candidate is against her admission as a privat-

dozent [instructor]. After all, the university senate is not a bathhouse.” She

was finally admitted to the faculty in 1919. A Jew, Noether fled Germany

in 1933 and joined the faculty at Bryn Mawr in the U.S.

In our example, the symmetry and the conservation law both

give the same information. Either statement can be satisfied only by

a circular orbit. That isn’t a coincidence. Physicist Emmy Noether

showed on very general mathematical grounds that for physical the-

ories of a certain type, every symmetry leads to a corresponding

conservation law. Although the precise formulation of Noether’s

theorem, and its proof, are too mathematical for this book, we’ll see

many examples like this one, in which the physical content of the

theorem is fairly straightforward.

The idea of perfect circular orbits seems very beautiful and in-

tuitively appealing. It came as a great disappointment, therefore,

when the astronomer Johannes Kepler discovered, by the painstak-

ing analysis of precise observations, that orbits such as the moon’s

were actually ellipses, not circles. This is the sort of thing that led

c / In

this

scene

from

Swan

the biologist Huxley to say, “The great tragedy of science is the slay-

Lake, the choreography has a

ing of a beautiful theory by an ugly fact.” The lesson of the story,

symmetry with respect to left and

then, is that symmetries are important and beautiful, but we can’t

right.

decide which symmetries are right based only on common sense or

aesthetics; their validity has to be determined based on observations

and experiments.

As a more modern example, consider the symmetry between

right and left. For example, we observe that a top spinning clockwise

has exactly the same behavior as a top spinning counterclockwise.

This kind of observation led physicists to believe, for hundreds of

years, that the laws of physics were perfectly symmetric with respect

to right and left.

This mirror symmetry appealed to physicists’

common sense. However, experiments by Chien-Shiung Wu et al. in

1957 showed that right-left symmetry was violated in certain types

of nuclear reactions. Physicists were thus forced to change their

d / C.S. Wu at Columbia Uni-

versity in 1963.

opinions about what constituted common sense.

8

Chapter 1

Conservation of Mass and Energy

1.2 Conservation of mass

We intuitively feel that matter shouldn’t appear or disappear out of

nowhere: that the amount of matter should be a conserved quan-

tity. If that was to happen, then it seems as though atoms would

have to be created or destroyed, which doesn’t happen in any phys-

ical processes that are familiar from everyday life, such as chemical

reactions. On the other hand, I’ve already cautioned you against

believing that a law of physics must be true just because it seems

appealing. The laws of physics have to be found by experiment, and

there seem to be experiments that are exceptions to the conserva-

tion of matter. A log weighs more than its ashes. Did some matter

simply disappear when the log was burned?

The French chemist Antoine-Laurent Lavoisier was the first sci-

entist to realize that there were no such exceptions. Lavoisier hy-

pothesized that when wood burns, for example, the supposed loss

of weight is actually accounted for by the escaping hot gases that

the flames are made of. Before Lavoisier, chemists had almost never

weighed their chemicals to quantify the amount of each substance

that was undergoing reactions. They also didn’t completely under-

stand that gases were just another state of matter, and hadn’t tried

performing reactions in sealed chambers to determine whether gases

were being consumed from or released into the air. For this they

had at least one practical excuse, which is that if you perform a gas-

releasing reaction in a sealed chamber with no room for expansion,

you get an explosion! Lavoisier invented a balance that was capable

e / Portrait of Monsieur Lavoisier

of measuring milligram masses, and figured out how to do reactions

and His Wife, by Jacques-Louis

in an upside-down bowl in a basin of water, so that the gases could

David, 1788. Lavoisier invented

expand by pushing out some of the water. In one crucial experi-

the concept of conservation of

ment, Lavoisier heated a red mercury compound, which we would

mass. The husband is depicted

with

his

scientific

apparatus,

now describe as mercury oxide (HgO), in such a sealed chamber.

while in the background on the

A gas was produced (Lavoisier later named it “oxygen”), driving

left is the portfolio belonging

out some of the water, and the red compound was transformed into

to Madame Lavoisier,

who is

silvery liquid mercury metal. The crucial point was that the total

thought to have been a student of

mass of the entire apparatus was exactly the same before and after

David’s.

the reaction. Based on many observations of this type, Lavoisier

proposed a general law of nature, that matter is always conserved.

self-check A

In ordinary speech, we say that you should “conserve” something, be-

cause if you don’t, pretty soon it will all be gone. How is this different

from the meaning of the term “conservation” in physics?

. Answer,

p. 179

Although Lavoisier was an honest and energetic public official,

he was caught up in the Terror and sentenced to death in 1794. He

requested a fifteen-day delay of his execution so that he could com-

plete some experiments that he thought might be of value to the

Republic. The judge, Coffinhal, infamously replied that “the state

Section 1.2

Conservation of mass

9

has no need of scientists.” As a scientific experiment, Lavoisier de-

cided to try to determine how long his consciousness would continue

after he was guillotined, by blinking his eyes for as long as possible.

He blinked twelve times after his head was chopped off. Ironically,

Judge Coffinhal was himself executed only three months later, falling

victim to the same chaos.

A stream of water

example 1

The stream of water is fatter near the mouth of the faucet, and

skinnier lower down. This can be understood using conservation

of mass. Since water is being neither created nor destroyed, the

mass of the water that leaves the faucet in one second must be

the same as the amount that flows past a lower point in the same

time interval. The water speeds up as it falls, so the two quan-

tities of water can only be equal if the stream is narrower at the

bottom.

Physicists are no different than plumbers or ballerinas in that

they have a technical vocabulary that allows them to make precise

distinctions. A pipe isn’t just a pipe, it’s a PVC pipe. A jump isn’t

f / Example 1.

just a jump, it’s a grand jeté. We need to be more precise now about

what we really mean by “the amount of matter,” which is what

we’re saying is conserved. Since physics is a mathematical science,

definitions in physics are usually definitions of numbers, and we

define these numbers operationally. An operational definition is one

that spells out the steps required in order to measure that quantity.

For example, one way that an electrician knows that current and

voltage are two different things is that she knows she has to do

completely different things in order to measure them with a meter.

If you ask a room full of ordinary people to define what is meant

by mass, they’ll probably propose a bunch of different, fuzzy ideas,

g

and speak as if they all pretty much meant the same thing: “how

/ The

time for one cycle of

vibration is related to the object’s

much space it takes up,” “how much it weighs,” “how much matter

mass.

is in it.” Of these, the first two can be disposed of easily. If we

were to define mass as a measure of how much space an object

occupied, then mass wouldn’t be conserved when we squished a

piece of foam rubber.

Although Lavoisier did use weight in his

experiments, weight also won’t quite work as the ultimate, rigorous

definition, because weight is a measure of how hard gravity pulls on

an object, and gravity varies in strength from place to place. Gravity

is measurably weaker on the top of a mountain that at sea level,

and much weaker on the moon. The reason this didn’t matter to

Lavoisier was that he was doing all his experiments in one location.

The third proposal is better, but how exactly should we define “how

h / Astronaut

Tamara

Jernigan

much matter?” To make it into an operational definition, we could

measures her mass aboard the

do something like figure g. A larger mass is harder to whip back

Space Shuttle. She is strapped

into a chair attached to a spring,

and forth — it’s harder to set into motion, and harder to stop once

like the mass in figure g. (NASA)

it’s started. For this reason, the vibration of the mass on the spring

will take a longer time if the mass is greater. If we put two different

10

Chapter 1

Conservation of Mass and Energy

masses on the spring, and they both take the same time to complete

one oscillation, we can define them as having the same mass.

Since I started this chapter by highlighting the relationship be-

tween conservation laws and symmetries, you’re probably wondering

what symmetry is related to conservation of mass. I’ll come back to

that at the end of the chapter.

When you learn about a new physical quantity, such as mass,

you need to know what units are used to measure it. This will lead

us to a brief digression on the metric system, after which we’ll come

back to physics.

1.3 Review of the metric system and

conversions

The metric system

Every country in the world besides the U.S. has adopted a sys-

tem of units known colloquially as the “metric system.” Even in

the U.S., the system is used universally by scientists, and also by

many engineers. This system is entirely decimal, thanks to the same

eminently logical people who brought about the French Revolution.

In deference to France, the system’s official name is the Système In-

ternational, or SI, meaning International System. (The phrase “SI

system” is therefore redundant.)

The metric system works with a single, consistent set of prefixes

(derived from Greek) that modify the basic units. Each prefix stands

for a power of ten, and has an abbreviation that can be combined

with the symbol for the unit. For instance, the meter is a unit of

distance. The prefix kilo- stands for 1000, so a kilometer, 1 km, is

a thousand meters.

In this book, we’ll be using a flavor of the metric system, the SI,

in which there are three basic units, measuring distance, time, and

mass. The basic unit of distance is the meter (m), the one for time

is the second (s), and for mass the kilogram (kg). Based on these

units, we can define others, e.g., m/s (meters per second) for the

speed of a car, or kg/s for the rate at which water flows through a

pipe. It might seem odd that we consider the basic unit of mass to

be the kilogram, rather than the gram. The reason for doing this

is that when we start defining other units starting from the basic

three, some of them come out to be a more convenient size for use

in everyday life. For example, there is a metric unit of force, the

newton (N), which is defined as the push or pull that would be able

to change a 1-kg object’s velocity by 1 m/s, if it acted on it for 1 s.

A newton turns out to be about the amount of force you’d use to

pick up your keys. If the system had been based on the gram instead

of the kilogram, then the newton would have been a thousand times

Section 1.3

Review of the metric system and conversions

11

smaller, something like the amount of force required in order to pick up a breadcrumb.

The following are the most common metric prefixes. You should

memorize them.

prefix

meaning

example

kilo-

k

1000

60 kg

= a person’s mass

centi-

c

1/100

28 cm

= height of a piece of paper

milli-

m

1/1000

1 ms

= time for one vibration of a guitar

string playing the note D

The prefix centi-, meaning 1/100, is only used in the centimeter;

a hundredth of a gram would not be written as 1 cg but as 10 mg.

The centi- prefix can be easily remembered because a cent is 1/100

of a dollar. The official SI abbreviation for seconds is “s” (not “sec”)

and grams are “g” (not “gm”).

You may also encounter the prefixes mega- (a million) and micro-

(one millionth).

Scientific notation

Most of the interesting phenomena in our universe are not on

the human scale. It would take about 1,000,000,000,000,000,000,000

bacteria to equal the mass of a human body. When the physicist

Thomas Young discovered that light was a wave, scientific notation

hadn’t been invented, and he was obliged to write that the time

required for one vibration of the wave was 1/500 of a millionth of

a millionth of a second. Scientific notation is a less awkward way

to write very large and very small numbers such as these. Here’s a

quick review.

Scientific notation means writing a number in terms of a product

of something from 1 to 10 and something else that is a power of ten.

For instance,

32 = 3.2 × 101

320 = 3.2 × 102

3200 = 3.2 × 103

. . .

Each number is ten times bigger than the last.

Since 101 is ten times smaller than 102 , it makes sense to use

the notation 100 to stand for one, the number that is in turn ten

times smaller than 101 . Continuing on, we can write 10−1 to stand

for 0.1, the number ten times smaller than 100 . Negative exponents

are used for small numbers:

3.2 = 3.2 × 100

0.32 = 3.2 × 10−1

0.032 = 3.2 × 10−2

. . .

12

Chapter 1

Conservation of Mass and Energy

A common source of confusion is the notation used on the dis-

plays of many calculators. Examples:

3.2 × 106

(written notation)

3.2E+6

(notation on some calculators)

3.26

(notation on some other calculators)

The last example is particularly unfortunate, because 3.26 really

stands for the number 3.2 × 3.2 × 3.2 × 3.2 × 3.2 × 3.2 = 1074, a

totally different number from 3.2 × 106 = 3200000. The calculator

notation should never be used in writing. It’s just a way for the

manufacturer to save money by making a simpler display.

self-check B

A student learns that 104 bacteria, standing in line to register for classes

at Paramecium Community College, would form a queue of this size:

The student concludes that 102 bacteria would form a line of this length:

Why is the student incorrect?

. Answer, p. 179

Conversions

I suggest you avoid memorizing lots of conversion factors be-

tween SI units and U.S. units. Suppose the United Nations sends

its black helicopters to invade California (after all who wouldn’t

rather live here than in New York City?), and institutes water flu-

oridation and the SI, making the use of inches and pounds into a

crime punishable by death. I think you could get by with only two

mental conversion factors:

1 inch = 2.54 cm

An object with a weight on Earth of 2.2 pounds-force has a

mass of 1 kg.

The first one is the present definition of the inch, so it’s exact. The

second one is not exact, but is good enough for most purposes. (U.S.

units of force and mass are confusing, so it’s a good thing they’re

not used in science. In U.S. units, the unit of force is the pound-

force, and the best unit to use for mass is the slug, which is about

14.6 kg.)

More important than memorizing conversion factors is under-

standing the right method for doing conversions. Even within the

SI, you may need to convert, say, from grams to kilograms. Differ-

ent people have different ways of thinking about conversions, but

the method I’ll describe here is systematic and easy to understand.

The idea is that if 1 kg and 1000 g represent the same mass, then

Section 1.3

Review of the metric system and conversions

13

we can consider a fraction like

103 g

1 kg

to be a way of expressing the number one. This may bother you. For

instance, if you type 1000/1 into your calculator, you will get 1000,

not one. Again, different people have different ways of thinking

about it, but the justification is that it helps us to do conversions,

and it works! Now if we want to convert 0.7 kg to units of grams,

we can multiply kg by the number one:

103 g

0.7 kg × 1 kg

If you’re willing to treat symbols such as “kg” as if they were vari-

ables as used in algebra (which they’re really not), you can then

cancel the kg on top with the kg on the bottom, resulting in

103 g

0.7 kg ×

= 700 g.

1 kg

To convert grams to kilograms, you would simply flip the fraction

upside down.

One advantage of this method is that it can easily be applied to

a series of conversions. For instance, to convert one year to units of

seconds,

365

days

24

60

min

60 s

1 year ×

hours

×

×

×

=

1 year

1

day

1

1

min

hour

= 3.15 × 107 s.

Should that exponent be positive or negative?

A common mistake is to write the conversion fraction incorrectly.

For instance the fraction

103 kg

(incorrect)

1 g

does not equal one, because 103 kg is the mass of a car, and 1 g is

the mass of a raisin. One correct way of setting up the conversion

factor would be

10−3 kg

(correct).

1 g

You can usually detect such a mistake if you take the time to check

your answer and see if it is reasonable.

If common sense doesn’t rule out either a positive or a negative

exponent, here’s another way to make sure you get it right. There

are big prefixes, like kilo-, and small ones, like milli-. In the example

above, we want the top of the fraction to be the same as the bottom.

Since k is a big prefix, we need to compensate by putting a small

number like 10−3 in front of it, not a big number like 103.

14

Chapter 1

Conservation of Mass and Energy

Discussion question

A

Each of the following conversions contains an error. In each case,

explain what the error is.

(a) 1000 kg × 1 kg = 1 g

1000 g

(b) 50 m × 1 cm = 0.5 cm

100 m

1.4 Conservation of energy

Energy

Consider the hockey puck in figure i. If we release it at rest, we

expect it to remain at rest. If it did start moving all by itself, that

would be strange: it would have to pick some direction in which

to move, and why would it pick that direction rather than some

other one? If we observed such a phenomenon, we would have to

conclude that that direction in space was somehow special. It would

be the favored direction in which hockey pucks (and presumably

other objects as well) preferred to move. That would violate our

intuition about the symmetry of space, and this is a case where our

i / A hockey puck is released

intuition is right: a vast number of experiments have all shown that

at rest.

If it spontaneously

that symmetry is a correct one. In other words, if you secretly pick

scooted off in some direction,

that would violate the symmetry

up the physics laboratory with a crane, and spin it around gently

of all directions in space.

with all the physicists inside, all their experiments will still come

out the same, regardless of the lab’s new orientation. If they don’t

have windows they can look out of, or any other external cues (like

the Earth’s magnetic field), then they won’t notice anything until

they hang up their lab coats for the evening and walk out into the

parking lot.

Another way of thinking about it is that a moving hockey puck

would have some energy, whereas a stationary one has none.

I

haven’t given you an operational definition of energy yet, but we’ll

gradually start to build one up, and it will end up fitting in pretty

well with your general idea of what energy means from everyday

life. Regardless of the mathematical details of how you would actu-

ally calculate the energy of a moving hockey puck, it makes sense

that a puck at rest has zero energy. It starts to look like energy is

conserved. A puck that initially has zero energy must continue to

have zero energy, so it can’t start moving all by itself.

You might conclude from this discussion that we have a new

example of Noether’s theorem: that the symmetry of space with re-

j / James

Joule

(1818-1889)

spect to different directions must be equivalent, in some mysterious

discovered the law of conserva-

way, to conservation of energy. Actually that’s not quite right, and

tion of energy.

the possible confusion is related to the fact that we’re not going

to deal with the full, precise mathematical statement of Noether’s

theorem. In fact, we’ll see soon that conservation of energy is re-

ally more closely related to a different symmetry, which is symmetry

Section 1.4

Conservation of energy

15

with respect to the passage of time.

The principle of inertia

Now there’s one very subtle thing about the example of the

hockey puck, which wouldn’t occur to most people. If we stand

on the ice and watch the puck, and we don’t see it moving, does

that mean that it really is at rest in some absolute sense? Remem-

ber, the planet earth spins once on its axis every 24 hours. At the

latitude where I live, this results in a speed of about 800 miles per

hour, or something like 400 meters per second. We could say, then

that the puck wasn’t really staying at rest. We could say that it

was really in motion at a speed of 400 m/s, and remained in motion

at that same speed. This may be inconsistent with our earlier de-

scription, but it is still consistent with the same description of the

laws of physics. Again, we don’t need to know the relevant formula

for energy in order to believe that if the puck keeps the same speed

(and its mass also stays the same), it’s maintaining the same energy.

In other words, we have two different frames of reference, both

equally valid. The person standing on the ice measures all velocities

relative to the ice, finds that the puck maintained a velocity of zero,

and says that energy was conserved. The astronaut watching the

scene from deep space might measure the velocities relative to her

own space station; in her frame of reference, the puck is moving at

400 m/s, but energy is still conserved.

This probably seems like common sense, but it wasn’t common

sense to one of the smartest people ever to live, the ancient Greek

philosopher Aristotle. He came up with an entire system of physics

k / Why

does

Aristotle

look

based on the premise that there is one frame of reference that is

so sad?

Is it because he’s

special: the frame of reference defined by the dirt under our feet.

realized that his entire system of

He believed that all motion had a tendency to slow down unless a

physics is wrong?

force was present to maintain it. Today, we know that Aristotle was

wrong. One thing he was missing was that he didn’t understand the

concept of friction as a force. If you kick a soccer ball, the reason

it eventually comes to rest on the grass isn’t that it “naturally”

wants to stop moving. The reason is that there’s a frictional force

from the grass that is slowing it down. (The energy of the ball’s

motion is transformed into other forms, such as heat and sound.)

Modern people may also have an easier time seeing his mistake,

because we have experience with smooth motion at high speeds.

For instance, consider a passenger on a jet plane who stands up

in the aisle and inadvertently drops his bag of peanuts. According

l / The jets are at rest.

The

to Aristotle, the bag would naturally slow to a stop, so it would

Empire State Building is moving.

become a life-threatening projectile in the cabin! From the modern

point of view, the cabin can just as well be considered to be at rest.

16

Chapter 1

Conservation of Mass and Energy

m / Galileo Galilei was the first physicist to state the principle of inertia (in

a somewhat different formulation than the one given here). His contradic-

tion of Aristotle had serious consequences. He was interrogated by the

Church authorities and convicted of teaching that the earth went around

the sun as a matter of fact and not, as he had promised previously, as a

mere mathematical hypothesis. He was placed under permanent house

arrest, and forbidden to write about or teach his theories. Immediately af-

ter being forced to recant his claim that the earth revolved around the sun,

the old man is said to have muttered defiantly “and yet it does move.”

The principle of inertia says, roughly, that all frames of reference

are equally valid:

The principle of inertia

The results of experiments don’t depend on the straight-line,

constant-speed motion of the apparatus.

Speaking slightly more precisely, the principle of inertia says that

if frame B moves at constant speed, in a straight line, relative to

frame A, then frame B is just as valid as frame A, and in fact an

observer in frame B will consider B to be at rest, and A to be moving.

The laws of physics will be valid in both frames. The necessity for

the more precise formulation becomes evident if you think about

examples in which the motion changes its speed or direction. For

instance, if you’re in a car that’s accelerating from rest, you feel

yourself being pressed back into your seat. That’s very different from

the experience of being in a car cruising at constant speed, which

produces no physical sensation at all. A more extreme example of

this is shown in figure n on page 18.

A frame of reference moving at constant speed in a straight line

is known as an inertial frame of reference. A frame that changes

its speed or direction of motion is called noninertial. The principle

of inertia applies only to inertial frames. The frame of reference

defined by an accelerating car is noninertial, but the one defined by

a car cruising at constant speed in a straight line is inertial.

Foucault’s pendulum

example 2

Earlier, I spoke as if a frame of reference attached to the surface

of the rotating earth was just as good as any other frame of ref-

erence. Now, with the more exact formulation of the principle of

inertia, we can see that that isn’t quite true. A point on the earth’s

surface moves in a circle, whereas the principle of inertia refers

only to motion in a straight line. However, the curve of the mo-

o / Foucault

demonstrates

tion is so gentle that under ordinary conditions we don’t notice

his pendulum to an audience at a

that the local dirt’s frame of reference isn’t quite inertial. The first

lecture in 1851.

demonstration of the noninertial nature of the earth-fixed frame of

reference was by L éon Foucault using a very massive pendulum

Section 1.4

Conservation of energy

17

n / This Air Force doctor volunteered to ride a rocket sled as a

medical experiment. The obvious effects on his head and face are not

because of the sled’s speed but because of its rapid changes in speed:

increasing in 2 and 3, and decreasing in 5 and 6. In 4 his speed is

greatest, but because his speed is not increasing or decreasing very

much at this moment, there is little effect on him.

(figure o) whose oscillations would persist for many hours with-

out becoming imperceptible. Although Foucault did his demon-

stration in Paris, it’s easier to imagine what would happen at the

north pole: the pendulum would keep swinging in the same plane,

but the earth would spin underneath it once every 24 hours. To

someone standing in the snow, it would appear that the pendu-

lum’s plane of motion was twisting. The effect at latitudes less

than 90 degrees turns out to be slower, but otherwise similar. The

Foucault pendulum was the first definitive experimental proof that

the earth really did spin on its axis, although scientists had been

convinced of its rotation for a century based on more indirect evi-

dence about the structure of the solar system.

People have a strong intuitive belief that there is a state of ab-

solute rest, and that the earth’s surface defines it. But Copernicus

proposed as a mathematical assumption, and Galileo argued as a

matter of physical reality, that the earth spins on its axis, and also

circles the sun. Galileo’s opponents objected that this was impossi-

ble, because we would observe the effects of the motion. They said,

for example, that if the earth was moving, then you would never

be able to jump up in the air and land in the same place again —

the earth would have moved out from under you. Galileo realized

18

Chapter 1

Conservation of Mass and Energy

that this wasn’t really an argument about the earth’s motion but

about physics. In one of his books, which were written in the form

of dialogues, he has the three characters debate what would happen

if a ship was cruising smoothly across a calm harbor and a sailor

climbed up to the top of its mast and dropped a rock. Would it hit

the deck at the base of the mast, or behind it because the ship had

moved out from under it? This is the kind of experiment referred to

in the principle of inertia, and Galileo knew that it would come out

the same regardless of the ship’s motion. His opponents’ reasoning,

as represented by the dialog’s stupid character Simplicio, was based

on the assumption that once the rock lost contact with the sailor’s

hand, it would naturally start to lose its forward motion. In other

words, they didn’t even believe in the idea that motion naturally

continues unless a force acts to stop it.

But the principle of inertia says more than that. It says that

motion isn’t even real: to a sailor standing on the deck of the ship,

the deck and the masts and the rigging are not even moving. People

on the shore can tell him that the ship and his own body are moving

in a straight line at constant speed. He can reply, “No, that’s an

illusion.

I’m at rest.

The only reason you think I’m moving is

because you and the sand and the water are moving in the opposite

direction.” The principle of inertia says that straight-line, constant-

speed motion is a matter of opinion. Thus things can’t “naturally”

slow down and stop moving, because we can’t even agree on which

things are moving and which are at rest.

If observers in different frames of reference disagree on velocities,

it’s natural to want to be able to convert back and forth. For motion

in one dimension, this can be done by simple addition.

A sailor running on the deck

example 3

. A sailor is running toward the front of a ship, and the other

sailors say that in their frame of reference, fixed to the deck, his

velocity is 7.0 m/s. The ship is moving at 1.3 m/s relative to the

shore. How fast does an observer on the beach say the sailor is

moving?

. They see the ship moving at 7.0 m/s, and the sailor moving even

faster than that because he’s running from the stern to the bow.

In one second, the ship moves 1.3 meters, but he moves 1.3 + 7.0

m, so his velocity relative to the beach is 8.3 m/s.

The only way to make this rule give consistent results is if we

define velocities in one direction as positive, and velocities in the

opposite direction as negative.

Running back toward the stern

example 4

. The sailor of example 3 turns around and runs back toward

the stern at the same speed relative to the deck. How do the

other sailors describe this velocity mathematically, and what do

Section 1.4

Conservation of energy

19

observers on the beach say?

. Since the other sailors described his original velocity as positive,

they have to call this negative. They say his velocity is now −7.0

m/s. A person on the shore says his velocity is 1.3+(−7.0) = −5.7

m/s.

Kinetic and gravitational energy

Now suppose we drop a rock. The rock is initially at rest, but

p / The

skater

has

con-

then begins moving. This seems to be a violation of conservation

verted

all

his

kinetic

energy

of energy, because a moving rock would have more energy. But ac-

into

gravitational

energy

on

tually this is a little like the example of the burning log that seems

the way up the side of the

pool.

Photo by J.D. Rogge,

to violate conservation of mass. Lavoisier realized that there was

www.sonic.net/∼shawn.

a second form of mass, the mass of the smoke, that wasn’t being

accounted for, and proved by experiments that mass was, after all,

conserved once the second form had been taken into account. In the

case of the falling rock, we have two forms of energy. The first is

the energy it has because it’s moving, known as kinetic energy. The

second form is a kind of energy that it has because it’s interacting

with the planet earth via gravity. This is known as gravitational en-

ergy.1 The earth and the rock attract each other gravitationally, and

the greater the distance between them, the greater the gravitational

energy — it’s a little like stretching a spring.

The SI unit of energy is the joule (J), and in those units, we find

that lifting a 1-kg mass through a height of 1 m requires 9.8 J of

energy. This number, 9.8 joules per meter per kilogram, is a measure

of the strength of the earth’s gravity near its surface. We notate this

number, known as the gravitational field, as g, and often round it

off to 10 for convenience in rough calculations. If you lift a 1-kg rock

to a height of 1 m above the ground, you’re giving up 9.8 J of the

energy you got from eating food, and changing it into gravitational

energy stored in the rock. If you then release the rock, it starts

transforming the energy into kinetic energy, until finally when the

rock is just about to hit the ground, all of that energy is in the form

of kinetic energy. That kinetic energy is then transformed into heat

q / As the skater free-falls, his

and sound when the rock hits the ground.

gravitational energy is converted

Stated in the language of algebra, the formula for gravitational

into kinetic energy.

energy is

GE = mgh,

where m is the mass of an object, g is the gravitational field, and h

is the object’s height.

A lever

example 5

Figure r shows two sisters on a seesaw. The one on the left

has twice as much mass, but she’s at half the distance from the

center. No energy input is needed in order to tip the seesaw. If

r / Example 5.

1You may also see this referred to in some books as gravitational potential

energy.

20

Chapter 1

Conservation of Mass and Energy

the girl on the left goes up a certain distance, her gravitational

energy will increase. At the same time, her sister on the right

will drop twice the distance, which results in an equal decrease in

energy, since her mass is half as much. In symbols, we have

(2m)gh

for the gravitational energy gained by the girl on the left, and

mg(2h)

for the energy lost by the one on the right. Both of these equal

2mgh, so the amounts gained and lost are the same, and energy

is conserved.

Looking at it another way, this can be thought of as an example

of the kind of experiment that you’d have to do in order to arrive

at the equation GE = mgh in the first place. If we didn’t already

know the equation, this experiment would make us suspect that

it involved the product mh, since that’s what’s the same for both

girls.

Once we have an equation for one form of energy, we can estab-

lish equations for other forms of energy. For example, if we drop a

rock and measure its final velocity, v, when it hits the ground, we

know how much GE it lost, so we know that’s how much KE it must

have had when it was at that final speed. Here are some imaginary

results from such an experiment.

m (kg)

v (m/s)

energy (J)

1.00

1.00

0.50

1.00

2.00

2.00

2.00

1.00

1.00

Comparing the first line with the second, we see that doubling

the object’s velocity doesn’t just double its energy, it quadruples it.

If we compare the first and third lines, however, we find that dou-

bling the mass only doubles the energy. This suggests that kinetic

energy is proportional to mass times the square of velocity, mv2,

and further experiments of this type would indeed establish such a

general rule. The proportionality factor equals 0.5 because of the

design of the metric system, so the kinetic energy of a moving object

is given by

1

KE =

mv2.

2

Energy in general

t / The

spinning

coin

slows

By this point, I’ve casually mentioned several forms of energy:

down. It looks like conservation

kinetic, gravitational, heat, and sound. This might be disconcert-

of energy is violated, but it isn’t.

ing, since we can get thoroughly messed up if we don’t realize that a

Section 1.4

Conservation of energy

21

s / A vivid demonstration that heat is a form of motion.

A small

amount of boiling water is poured into the empty can, which rapidly fills

up with hot steam. The can is then sealed tightly, and soon crumples.

This can be explained as follows. The high temperature of the steam is

interpreted as a high average speed of random motions of its molecules.

Before the lid was put on the can, the rapidly moving steam molecules

pushed their way out of the can, forcing the slower air molecules out of

the way. As the steam inside the can thinned out, a stable situation was

soon achieved, in which the force from the less dense steam molecules

moving at high speed balanced against the force from the more dense but

slower air molecules outside. The cap was put on, and after a while the

steam inside the can reached the same temperature as the air outside.

The force from the cool, thin steam no longer matched the force from the

cool, dense air outside, and the imbalance of forces crushed the can.

certain form of energy is important in a particular situation. For in-

stance, the spinning coin in figure t gradually loses its kinetic energy,

and we might think that conservation of energy was therefore being

violated. However, whenever two surfaces rub together, friction acts

to create heat. The correct analysis is that the coin’s kinetic energy

is gradually converted into heat.

One way of making the proliferation of forms of energy seem less

scary is to realize that many forms of energy that seem different on

the surface are in fact the same. One important example is that

heat is actually the kinetic energy of molecules in random motion,

so where we thought we had two forms of energy, in fact there is

only one. Sound is also a form of kinetic energy: it’s the vibration

of air molecules.

This kind of unification of different types of energy has been a

process that has been going on in physics for a long time, and at

this point we’ve gotten it down the point where there really only

appear to be four forms of energy:

22

Chapter 1

Conservation of Mass and Energy

1. kinetic energy

2. gravitational energy

3. electrical energy

4. nuclear energy

We don’t even encounter nuclear energy in everyday life (except in

the sense that sunlight originates as nuclear energy), so really for

most purposes the list only has three items on it. Of these three,

electrical energy is the only form that we haven’t talked about yet.

The interactions between atoms are all electrical, so this form of

energy is what’s responsible for all of chemistry. The energy in the

food you eat, or in a tank of gasoline, are forms of electrical energy.

You take the high road and I’ll take the low road.

example 6

. Figure u shows two ramps which two balls will roll down. Com-

pare their final speeds, when they reach point B. Assume friction

is negligible.

. Each ball loses some gravitational energy because of its de-

creasing height above the earth, and conservation of energy says

that it must gain an equal amount of kinetic energy (minus a lit-

tle heat created by friction). The balls lose the same amount of

height, so their final speeds must be equal.

The birth of stars

example 7

u / Example 6.

Orion is the easiest constellation to find. You can see it in the

winter, even if you live under the light-polluted skies of a big city.

Figure v shows an interesting feature of this part of the sky that

you can easily pick out with an ordinary camera (that’s how I took

the picture) or a pair of binoculars. The three stars at the top are

Orion’s belt, and the stuff near the lower left corner of the picture

is known as his sword — to the naked eye, it just looks like three

more stars that aren’t as bright as the stars in the belt. The mid-

dle “star” of the sword, however, isn’t a star at all. It’s a cloud

of gas, known as the Orion Nebula, that’s in the process of col-

lapsing due to gravity. Like the pool skater on his way down, the

gas is losing gravitational energy. The results are very different,

however. The skateboard is designed to be a low-friction device,

so nearly all of the lost gravitational energy is converted to ki-

netic energy, and very little to heat. The gases in the nebula flow

and rub against each other, however, so most of the gravitational

energy is converted to heat. This is the process by which stars

v / Example 7.

are born: eventually the core of the gas cloud gets hot enough to

ignite nuclear reactions.

Section 1.4

Conservation of energy

23

Lifting a weight

example 8

. At the gym, you lift a mass of 40 kg through a height of 0.5

m. How much gravitational energy is required? Where does this

energy come from?

. The strength of the gravitational field is 10 joules per kilogram

per meter, so after you lift the weight, its gravitational energy will

be greater by 10 × 40 × 0.5 = 200 joules.

Energy is conserved, so if the weight gains gravitational energy,

something else somewhere in the universe must have lost some.

The energy that was used up was the energy in your body, which

came from the food you’d eaten. This is what we refer to as

“burning calories,” since calories are the units normally used to

describe the energy in food, rather than metric units of joules.

In fact, your body uses up even more than 200 J of food energy,

because it’s not very efficient. The rest of the energy goes into

heat, which is why you’ll need a shower after you work out. We

can summarize this as

food energy → gravitational energy + heat.

Lowering a weight

example 9

. After lifting the weight, you need to lower it again. What’s hap-

pening in terms of energy?

w / Example 10.

. Your body isn’t capable of accepting the energy and putting it

back into storage. The gravitational energy all goes into heat.

(There’s nothing fundamental in the laws of physics that forbids

this. Electric cars can do it — when you stop at a stop sign, the

car’s kinetic energy is absorbed back into the battery, through a

generator.)

Absorption and emission of light

example 10

Light has energy. Light can be absorbed by matter and trans-

formed into heat, but the reverse is also possible: an object can

glow, transforming some of its heat energy into light. Very hot ob-

jects, like a candle flame or a welding torch, will glow in the visible

part of the spectrum, as in figure w.

Objects at lower temperatures will also emit light, but in the in-

frared part of the spectrum, i.e., the part of the rainbow lying

beyond the red end, which humans can’t see. The photos in

figure x were taken using a camera that is sensitive to infrared

light. The cyclist locked his rear brakes suddenly, and skidded

to a stop. The kinetic energy of the bike and his body are rapidly

transformed into heat by the friction between the tire and the floor.

x / Example 10.

In the first panel, you can see the glow of the heated strip on the

floor, and in the second panel, the heated part of the tire.

24

Chapter 1

Conservation of Mass and Energy

Heavy objects don’t fall faster

example 11

Stand up now, take off your shoe, and drop it alongside a much

less massive object such as a coin or the cap from your pen.

Did that surprise you? You found that they both hit the ground

at the same time. Aristotle wrote that heavier objects fall faster

than lighter ones. He was wrong, but Europeans believed him for

thousands of years, partly because experiments weren’t an ac-

cepted way of learning the truth, and partly because the Catholic

Church gave him its posthumous seal of approval as its official

philosopher.

Heavy objects and light objects have to fall the same way, be-

cause conservation laws are additive — we find the total energy

of an object by adding up the energies of all its atoms. If a single

atom falls through a height of one meter, it loses a certain amount

of gravitational energy and gains a corresponding amount of ki-

netic energy. Kinetic energy relates to speed, so that determines

how fast it’s moving at the end of its one-meter drop. (The same

reasoning could be applied to any point along the way between

zero meters and one.)

Now what if we stick two atoms together? The pair has double

the mass, so the amount of gravitational energy transformed into

kinetic energy is twice as much. But twice as much kinetic energy

is exactly what we need if the pair of atoms is to have the same

speed as the single atom did. Continuing this train of thought, it

doesn’t matter how many atoms an object contains; it will have

the same speed as any other object after dropping through the

same height.

1.5 Newton’s law of gravity

Why does the gravitational field on our planet have the particular

value it does? For insight, let’s compare with the strength of gravity

elsewhere in the universe:

location

g (joules per kg per m)

asteroid Vesta (surface)

0.3

earth’s moon (surface)

1.6

Mars (surface)

3.7

earth (surface)

9.8

Jupiter (cloud-tops)

26

sun (visible surface)

270

typical neutron star (surface)

1012

black hole (center)

infinite according to some

theories, on the order of

1052 according to others

A good comparison is Vesta versus a neutron star.

They’re

roughly the same size, but they have vastly different masses — a

Section 1.5

Newton’s law of gravity

25

teaspoonful of neutron star matter would weigh a million tons! The

different mass must be the reason for the vastly different gravita-

tional fields. (The notation 1012 means 1 followed by 12 zeroes.)

This makes sense, because gravity is an attraction between things

that have mass.

The mass of an object, however, isn’t the only thing that deter-

mines the strength of its gravitational field, as demonstrated by the

difference between the fields of the sun and a neutron star, despite

their similar masses. The other variable that matters is distance.

Because a neutron star’s mass is compressed into such a small space

(comparable to the size of a city), a point on its surface is within a

fairly short distance from every part of the star. If you visited the

surface of the sun, however, you’d be millions of miles away from

most of its atoms.

As a less exotic example, if you travel from the seaport of Guaya-

quil, Ecuador, to the top of nearby Mt. Cotopaxi, you’ll experience

a slight reduction in gravity, from 9.7806 to 9.7624 J/kg/m. This is

because you’ve gotten a little farther from the planet’s mass. Such

differences in the strength of gravity between one location and an-

other on the earth’s surface were first discovered because pendulum

clocks that were correctly calibrated in one country were found to

run too fast or too slow when they were shipped to another location.

The general equation for an object’s gravitational field was dis-

covered by Isaac Newton, by working backwards from the observed

motion of the planets:2

GM

g =

,

d2

where M is the mass of the object, d is the distance from the object,

and G is a constant that is the same everywhere in the universe. This

is known as Newton’s law of gravity.3 This type of relationship, in

which an effect is inversely proportional to the square of the distance

from the object creating the effect, is known as an inverse square

law. For example, the intensity of the light from a candle obeys an

inverse square law, as discussed in subsection 7.2.1 on page 140.

self-check C

Mars is about twice as far from the sun as Venus. Compare the strength

of the sun’s gravitational field as experienced by Mars with the strength

of the field felt by Venus.

. Answer, p. 179

Newton’s law of gravity really gives the field of an individual

atom, and the field of a many-atom object is the sum of the fields

of the atoms. Newton was able to prove mathematically that this

scary sum has an unexpectedly simple result in the case of a spherical

object such as a planet: the result is the same as if all the object’s

2Example 12 on page 50 shows the type of reasoning that Newton had to go

y / Isaac Newton (1642-1727)

through.

3This is not the form in which Newton originally wrote the equation.

26

Chapter 1

Conservation of Mass and Energy

mass had been concentrated at its center.

Newton showed that his theory of gravity could explain the or-

bits of the planets, and also finished the project begun by Galileo

of driving a stake through the heart of Aristotelian physics. His

book on the motion of material objects, the Mathematical Princi-

ples of Natural Philosophy, was uncontradicted by experiment for

200 years, but his other main work, Optics, was on the wrong track

due to his conviction that light was composed of particles rather

than waves. He was an avid alchemist, an embarrassing fact that

modern scientists would like to forget. Newton was on the winning

side of the revolution that replaced King James II with William and

Mary of Orange, which led to a lucrative post running the English

royal mint; he worked hard at what could have been a sinecure, and

took great satisfaction from catching and executing counterfeiters.

Newton’s personal life was less happy, as we’ll see in chapter 5.

Newton’s apple

example 12

A charming legend attested to by Newton’s niece is that he first

conceived of gravity as a universal attraction after seeing an ap-

ple fall from a tree. He wondered whether the force that made the

apple fall was the same one that made the moon circle the earth

rather than flying off straight. Newton had astronomical data that

allowed him to calculate that the gravitational field the moon ex-

perienced from the earth was 1/3600 as strong as the field on

the surface of the earth.4 (The moon has its own gravitational

field, but that’s not what we’re talking about.) The moon’s dis-

tance from the earth is 60 times greater than the earth’s radius,

so this fit perfectly with an inverse-square law: 60 × 60 = 3600.

1.6 Noether’s theorem for energy

Now we’re ready for our first full-fledged example of Noether’s the-

orem. Conservation of energy is a law of physics, and Noether’s

theorem says that the laws of physics come from symmetry. Specif-

ically, Noether’s theorem says that every symmetry implies a con-

servation law. Conservation of energy comes from a symmetry that

we haven’t even discussed yet, but one that is simple and intuitively

appealing: as time goes by, the universe doesn’t change the way it

works. We’ll call this time symmetry.

We have strong evidence for time symmetry, because when we

see a distant galaxy through a telescope, we’re seeing light that

has taken billions of years to get here. A telescope, then, is like a

time machine. For all we know, alien astronomers with advanced

technology may be observing our planet right now,5 but if so, they’re

4See example 12 on page 50.

5

z / Example 12.

Our present technology isn’t good enough to let us pick the planets of other

solar systems out from the glare of their suns, except in a few exceptional cases.

Section 1.6

Noether’s theorem for energy

27

seeing it not as it is now but as it was in the distant past, perhaps in the age of the dinosaurs, or before life even evolved here. As we

observe a particularly distant, and therefore ancient, supernova, we

see that its explosion plays out in exactly the same way as those

that are closer, and therefore more recent.

Now suppose physics really does change from year to year, like

politics, pop music, and hemlines. Imagine, for example, that the

“constant” G in Newton’s law of gravity isn’t quite so constant. One

day you might wake up and find that you’ve lost a lot of weight with-

out dieting or exercise, simply because gravity has gotten weaker

since the day before.

If you know about such changes in G over time, it’s the ultimate

insider information. You can use it to get as rich as Croesus, or even

Bill Gates. On a day when G is low, you pay for the energy needed

to lift a large mass up high. Then, on a day when gravity is stronger,

you lower the mass back down, extracting its gravitational energy.

The key is that the energy you get back out is greater than what

you originally had to put in. You can run the cycle over and over

again, always raising the weight when gravity is weak, and lowering

it when gravity is strong. Each time, you make a profit in energy.

Everyone else thinks energy is conserved, but your secret technique

allows you to keep on increasing and increasing the amount of energy

in the universe (and the amount of money in your bank account).

The scheme can be made to work if anything about physics

changes over time, not just gravity. For instance, suppose that the

mass of an electron had one value today, and a slightly different

value tomorrow. Electrons are one of the basic particles from which

atoms are built, so on a day when the mass of electrons is low, every

physical object has a slightly lower mass. In problem 14 on page

35, you’ll work out a way that this could be used to manufacture

energy out of nowhere.

Sorry, but it won’t work.

Experiments show that G doesn’t

change measurably over time, nor does there seem to be any time

variation in any of the other rules by which the universe works.6 If

archaeologists find a copy of this book thousands of years from now,

they’ll be able to reproduce all the experiments you’re doing in this

course.

I’ve probably convinced you that if time symmetry was violated,

then conservation of energy wouldn’t hold. But does it work the

6In 2002, there have been some reports that the properties of atoms as ob-

served in distant galaxies are slightly different than those of atoms here and now.

If so, then time symmetry is weakly violated, and so is conservation of energy.

However, this is a revolutionary claim, and it needs to be examined carefully.

The change being claimed is large enough that, if it’s real, it should be detectable from one year to the next in ultra-high-precision laboratory experiments here on

earth.

28

Chapter 1

Conservation of Mass and Energy

other way around? If time symmetry is valid, must there be a law

of conservation of energy? Logically, that’s a different question. We

may be able to prove that if A is false, then B must be false, but

that doesn’t mean that if A is true, B must be true as well. For

instance, if you’re not a criminal, then you’re presumably not in

jail, but just because someone is a criminal, that doesn’t mean he

is in jail — some criminals never get caught.

Noether’s theorem does work the other way around as well: if

physics has a certain symmetry, then there must be a certain cor-

responding conservation law. This is a stronger statement. The

full-strength version of Noether’s theorem can’t be proved without

a model of light and matter more detailed than the one currently at

our disposal.

1.7 Equivalence of mass and energy

Mass-energy

You’ve encountered two conservation laws so far: conservation

of mass and conservation of energy. If conservation of energy is a

consequence of symmetry, is there a deeper reason for conservation

of mass?

Actually they’re not even separate conservation laws. Albert

Einstein found, as a consequence of his theory of relativity, that

mass and energy are equivalent, and are not separately conserved

— one can be converted into the other. Imagine that a magician

waves his wand, and changes a bowl of dirt into a bowl of lettuce.

You’d be impressed, because you were expecting that both dirt and

lettuce would be conserved quantities. Neither one can be made to

vanish, or to appear out of thin air. However, there are processes

that can change one into the other. A farmer changes dirt into

lettuce, and a compost heap changes lettuce into dirt. At the most

fundamental level, lettuce and dirt aren’t really different things at

all; they’re just collections of the same kinds of atoms — carbon,

hydrogen, and so on.

We won’t examine relativity in detail in this book, but mass-

energy equivalence is an inevitable implication of the theory, and

it’s the only part of the theory that most people have heard of, via

the famous equation E = mc2. This equation tells us how much

energy is equivalent to how much mass: the conversion factor is the

square of the speed of light, c. Since c is a big number, you get

a really really big number when you multiply it by itself to get c2.

This means that even a small amount of mass is equivalent to a very

large amount of energy.

Section 1.7

Equivalence of mass and energy

29

aa / Example 13.

Gravity bending light

example 13

Gravity is a universal attraction between things that have mass,

and since the energy in a beam of light is equivalent to some very

small amount of mass, we expect that light will be affected by

gravity, although the effect should be very small. The first exper-

imental confirmation of relativity came in 1919 when stars next

to the sun during a solar eclipse were observed to have shifted

a little from their ordinary position. (If there was no eclipse, the

glare of the sun would prevent the stars from being observed.)

Starlight had been deflected by the sun’s gravity. Figure aa is a

photographic negative, so the circle that appears bright is actually

the dark face of the moon, and the dark area is really the bright

corona of the sun. The stars, marked by lines above and below

them, appeared at positions slightly different than their normal

ones.

Black holes

example 14

A star with sufficiently strong gravity can prevent light from leav-

ing. Quite a few black holes have been detected via their gravita-

tional forces on neighboring stars or clouds of gas and dust.

ab / A New York Times head-

line from November 10, 1919,

Because mass and energy are like two different sides of the same

describing

the

observations

coin, we may speak of mass-energy, a single conserved quantity,

discussed in example 13.

found by adding up all the mass and energy, with the appropriate

conversion factor: E + mc2.

30

Chapter 1

Conservation of Mass and Energy

A rusting nail

example 15

. An iron nail is left in a cup of water until it turns entirely to rust.

The energy released is about 500,000 joules. In theory, would a

sufficiently precise scale register a change in mass? If so, how

much?

. The energy will appear as heat, which will be lost to the envi-

ronment. The total mass-energy of the cup, water, and iron will

indeed be lessened by 500,000 joules. (If it had been perfectly

insulated, there would have been no change, since the heat en-

ergy would have been trapped in the cup.) The speed of light in

metric units is c = 3 × 108 meters per second (scientific notation

for 3 followed by 8 zeroes), so converting to mass units, we have

E

m = c2

500, 000

=

3 × 1082

= 0.000000000006 kilograms.

(The design of the metric system is based on the meter, the kilo-

gram, and the second. The joule is designed to fit into this sys-

tem, so the result comes out in units of kilograms.) The change in

mass is too small to measure with any practical technique. This is

because the square of the speed of light is such a large number

in metric units.

The correspondence principle

The realization that mass and energy are not separately con-

served is our first example of a general idea called the correspondence

principle. When Einstein came up with relativity, conservation of

energy had been accepted by physicists for decades, and conserva-

tion of mass for over a hundred years.

Does an example like this mean that physicists don’t know what

they’re talking about? There is a recent tendency among social sci-

entists to deny that the scientific method even exists, claiming that

science is no more than a social system that determines what ideas

to accept based on an in-group’s criteria. If science is an arbitrary

social ritual, it would seem difficult to explain its effectiveness in

building such useful items as airplanes, CD players and sewers. If

voodoo and astrology were no less scientific in their methods than

chemistry and physics, what was it that kept them from producing

anything useful? This silly attitude was effectively skewered in a

famous hoax carried out in 1996 by New York University physicist

Alan Sokal. Sokal wrote an article titled “Transgressing the Bound-

aries: Toward a Transformative Hermeneutics of Quantum Gravity,”

and got it accepted by a cultural studies journal called Social Text.7

7The paper appeared in Social Text

#46/47

(1996)

pp.

217-

Section 1.7

Equivalence of mass and energy

31

The scientific content of the paper is a carefully constructed soup of mumbo jumbo, using technical terms to create maximum confusion;

I can’t make heads or tails of it, and I assume the editors and peer

reviewers at Social Text understood even less. The physics, however,

is mixed in with cultural relativist statements designed to appeal to

them — “. . . the truth claims of science are inherently theory-laden

and self-referential” — and footnoted references to academic articles

such as “Irigaray’s and Hayles’ exegeses of gender encoding in fluid

mechanics . . . and . . . Harding’s comprehensive critique of the gen-

der ideology underlying the natural sciences in general and physics

in particular. . . ” On the day the article came out, Sokal published

a letter explaining that the whole thing had been a parody — one

that apparently went over the heads of the editors of Social Text.

What keeps physics from being merely a matter of fashion is that

it has to agree with experiments and observations. If a theory such

as conservation of mass or conservation of energy became accepted

in physics, it was because it was supported by a vast number of ex-

periments. It’s just that experiments never have perfect accuracy, so

a discrepancy such as the tiny change in the mass of the rusting nail

in example 15 was undetectable. The old experiments weren’t all

wrong. They were right, within their limitations. If someone comes

along with a new theory he claims is better, it must still be consis-

tent with all the same experiments. In computer jargon, it must be

backward-compatible. This is called the correspondence principle:

new theories must be compatible with old ones in situations where

they are both applicable. The correspondence principle tells us that

we can still use an old theory within the realm where it works, so

for instance I’ll typically refer to conservation of mass and conserva-

tion of energy in this book rather than conservation of mass-energy,

except in cases where the new theory is actually necessary.

Ironically, the extreme cultural relativists want to attack what

they see as physical scientists’ arrogant claims to absolute truth, but

what they fail to understand is that science only claims to be able

to find partial, provisional truth. The correspondence principle tells

us that each of today’s scientific truths can be superseded tomorrow

by another truth that is more accurate and more broadly applicable.

It also tells us that today’s truth will not lose any value when that

happens.

252.

The

full

text

is

available

on

Professor

Sokal’s

web

page

at

www.physics.nyu.edu/faculty/sokal/.

32

Chapter 1

Conservation of Mass and Energy

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

Convert 134 mg to units of kg, writing your answer in scientific

notation.

. Solution, p. 180

2

Compute the following things. If they don’t make sense be-

cause of units, say so.

(a) 3 cm + 5 cm

(b) 1.11 m + 22 cm

(c) 120 miles + 2.0 hours

(d) 120 miles / 2.0 hours

3

Your backyard has brick walls on both ends. You measure a

distance of 23.4 m from the inside of one wall to the inside of the

other. Each wall is 29.4 cm thick. How far is it from the outside

of one wall to the outside of the other? Pay attention to significant

figures.

4

The speed of light is 3.0 × 108 m/s. Convert this to furlongs

per fortnight. A furlong is 220 yards, and a fortnight is 14 days. An

√

inch is 2.54 cm.

5

Express each of the following quantities in micrograms: √

(a) 10 mg, (b) 104 g, (c) 10 kg, (d) 100 × 103 g, (e) 1000 ng.

6

In the last century, the average age of the onset of puberty for

girls has decreased by several years. Urban folklore has it that this

is because of hormones fed to beef cattle, but it is more likely to be

because modern girls have more body fat on the average and pos-

sibly because of estrogen-mimicking chemicals in the environment

from the breakdown of pesticides. A hamburger from a hormone-

implanted steer has about 0.2 ng of estrogen (about double the

amount of natural beef).

A serving of peas contains about 300

ng of estrogen. An adult woman produces about 0.5 mg of estrogen

per day (note the different unit!). (a) How many hamburgers would

a girl have to eat in one day to consume as much estrogen as an

adult woman’s daily production? (b) How many servings of peas?

√

7

You jump up straight up in the air. When do you have the

greatest gravitational energy? The greatest kinetic energy? (Based

on a problem by Serway and Faughn.)

8

Anya and Ivan lean over a balcony side by side. Anya throws a

penny downward with an initial speed of 5 m/s. Ivan throws a penny

upward with the same speed. Both pennies end up on the ground

below. Compare their kinetic energies and velocities on impact.

Problems

33

9

(a) If weight B moves down by a certain amount, how much

does weight A move up or down?

(b) What should the ratio of the two weights be if they are to bal-

ance? Explain in terms of conservation of energy.

10

(a) You release a magnet on a tabletop near a big piece of

iron, and the magnet leaps across the table to the iron. Does the

magnetic energy increase, or decrease? Explain.

(b) Suppose instead that you have two repelling magnets. You give

them an initial push towards each other, so they decelerate while

approaching each other. Does the magnetic energy increase, or de-

Problem 9.

crease? Explain.

11

For an astronaut sealed inside a space suit, getting rid of body

heat can be difficult. Suppose an astronaut is performing vigorous

physical activity, expending 200 watts of power. An energy of 200

kJ is enough to raise her body temperature by 1◦C. If none of the

heat can escape from her space suit, how long will it take before her

body temperature rises by 6◦C (11◦F), an amount sufficient to kill

√

her? Express your answer in units of minutes.

Problem 12.

12

The multiflash photograph below shows a collision between

two pool balls. The ball that was initially at rest shows up as a dark

image in its initial position, because its image was exposed several

times before it was struck and began moving.

By making mea-

surements on the figure, determine whether or not energy appears

to have been conserved in the collision. What systematic effects

would limit the accuracy of your test? (From an example in PSSC

Physics.)

34

Chapter 1

Conservation of Mass and Energy

13

How high above the surface of the earth should a rocket be

in order to have 1/100 of its normal weight? Express your answer

√

in units of earth radii.

14

As suggested on page 28, imagine that the mass of the electron

rises and falls over time. (Since all electrons are identical, physicists

generally talk about “the electron” collectively, as in “the modern

man wants more than just beer and sports.”)

The idea is that

all electrons are increasing and decreasing their masses in unison,

and at any given time, they’re all identical. They’re like a litter

of puppies whose weights are all identical on any given day, but

who all change their weights in unison from one month to the next.

Suppose you were the only person who knew about these small day-

to-day changes in the mass of the electron. Find a plan for violating

conservation of energy and getting rich.

15

A typical balance like the ones used in school classes can be

read to an accuracy of about plus or minus 0.1 grams, or 10−4 kg.

What if the laws of physics had been designed around a different

value of the speed of light? To make mass-energy equivalence de-

tectable in example 15 on page 31 using an ordinary balance, would

c have to be smaller than it is in our universe, or bigger? Find the

√

value of c for which the effect would be just barely detectable.

16

(a) A free neutron (as opposed to a neutron bound into an

atomic nucleus) is unstable, and decays radioactively into a proton,

an electron, and a particle called an antineutrino, which fly off in

three different directions. The masses are as follows:

neutron

1.67495 × 10−27 kg

proton

1.67265 × 10−27 kg

electron

0.00091 × 10−27 kg

antineutrino

negligible

√

Find the energy released in the decay of a free neutron.

(b) Neutrons and protons make up essentially all of the mass of the

ordinary matter around us. We observe that the universe around us

has no free neutrons, but lots of free protons (the nuclei of hydrogen,

which is the element that 90% of the universe is made of). We find

neutrons only inside nuclei along with other neutrons and protons,

not on their own.

If there are processes that can convert neutrons into protons, we

might imagine that there could also be proton-to-neutron conver-

sions, and indeed such a process does occur sometimes in nuclei

that contain both neutrons and protons: a proton can decay into a

neutron, a positron, and a neutrino. A positron is a particle with

the same properties as an electron, except that its electrical charge

is positive (see chapter 5). A neutrino, like an antineutrino, has

negligible mass.

Although such a process can occur within a nucleus, explain why

Problems

35

it cannot happen to a free proton. (If it could, hydrogen would be

radioactive, and you wouldn’t exist!)

17

(a) A 1.0 kg rock is released from rest, and drops 1.0 m. Find

√

the amount of gravitational energy released.

√

(b) Find the rock’s kinetic energy at the end of its fall.

√

(c) Find the rock’s velocity at the end of its fall.

36

Chapter 1

Conservation of Mass and Energy

Chapter 2

Conservation of

Momentum

Fantasy novelist T.H. White invented a wonderful phrase that has

since entered into popular culture: “Everything not forbidden is

compulsory.” Originally intended as a satire of totalitarianism, it

was taken up by physicist Murray Gell-Mann as a metaphor for

physics. What he meant was that the laws of physics forbid all the

impossible things, and what’s left over is what really happens. Con-

servation of mass and energy prevent many things from happening.

Objects can’t disappear into thin air, and you can’t run your car

forever without putting gas in it.

Some other processes are impossible, but not forbidden by these

two conservation laws. In the martial arts movie Crouching Tiger,

Hidden Dragon, those who have received mystical enlightenment are

able to violate the laws of physics. Some of the violation, such as

their ability to fly, are obvious, but others are a little more subtle.

The rebellious young heroine/antiheroine Jen Yu gets into an argu-

ment while sitting at a table in a restaurant. A young tough, Iron

Arm Lu, comes running toward her at full speed, and she puts up one

arm and effortlessly makes him bounce back, without even getting

out of her seat or bracing herself against anything. She does all this

between bites. It’s impossible, but how do we know it’s impossible?

It doesn’t violate conservation of mass, because neither character’s

mass changes. It conserves energy as well, since the rebounding Lu

has the same energy he started with.

Suppose you live in a country where the only laws are prohibi-

37

tions against murder and robbery. One day someone covers your

house with graffiti, and the authorities refuse to prosecute, because

no crime was committed. You’re convinced of the need for a new

law against vandalism. Similarly, the story of Jen Yu and Iron Arm

Lu shows that we need a new conservation law.

2.1 Translation symmetry

The most fundamental laws of physics are conservation laws, and

Noether’s theorem tells us that conservation laws are the way they

are because of symmetry. Time symmetry is responsible for con-

servation of energy, but time is like a river with only two direc-

tions, past and future. What’s impossible about Lu’s motion is the

abrupt reversal in the direction of his motion in space, but neither

time symmetry nor energy conservation tell us anything about di-

rections in space. When you put gas in your car, you don’t have

to decide whether you want to buy north gas or south gas, east,

west, up or down gas. Energy has no direction. What we need is

a new conserved quantity that has a direction in space, and such

a conservation law can only come from a symmetry that relates

to space. Since we’ve already had some luck with time symmetry,

which says that the laws of physics are the same at all times, it

seems reasonable to turn now to the possibility of a new type of

symmetry, which would state that the laws of physics are the same

in all places in space. This is known as translation symmetry, where

the word “translation” is being used in a mathematical sense that

means sliding something around without rotating it.

Translation symmetry would seem reasonable to most people,

but you’ll see that it ends up producing some very surprising results.

To see how, it will be helpful to imagine the consequences of a

violation of translation symmetry. What if, like the laws of nations,

the laws of physics were different in different places? What would

happen, and how would we detect it? We could try doing the same

experiment in two different places and comparing the results, but

it’s even easier than that. Tap your finger on this spot on the page

×

and then wait a second and do it again. Did both taps occur at

the same point in space? You’re probably thinking that’s a silly

question; am I just checking whether you followed my directions?

Not at all. Consider the whole scene from the point of view of a

Martian who is observing it through a powerful telescope from her

home planet. (You didn’t draw the curtains, did you?) From her

point of view, the earth is spinning on its axis and orbiting the sun,

at speeds measured in thousands of kilometers per hour. According

to her, your second finger tap happened at a point in space about 30

kilometers from the first. If you want to impress the Martians and

38

Chapter 2

Conservation of Momentum

win the Martian version of the Nobel Prize for detecting a violation of translation symmetry, all you have to do is perform a physics

experiment twice in the same laboratory, and show that the result

is different.

But who’s to say that the Martian point of view is the right one?

It gets a little thorny now. How do you know that what you detected

was a violation of translation symmetry at all? Maybe it was just

a violation of time symmetry. The Martian Nobel committee isn’t

going to give you the prize based on an experiment this ambiguous.

A possible scheme for resolving the ambiguity would be to wait a

year and do the same experiment a third time. After a year, the

earth will have completed one full orbit around the sun, and your

lab will be back in the same spot in space. If the third experiment

comes out the same as the first one, then you can make a strong

argument that what you’ve detected is an asymmetry of space, not

time. There’s a problem, however. You and the Martians agree that

the earth is back in the same place after a year, but what about an

observer from another solar system, whose planet orbits a different

star? This observer says that our whole solar system is in motion.

To him, the earth’s motion around our sun looks like a spiral or a

corkscrew, since the sun is itself moving.

2.2 The principle of inertia

Symmetry and inertia

This story shows that translation symmetry is closely related to

the relative nature of motion, as expressed by the principle of inertia.

Riding in a train on a long, straight track at constant speed, how

can you even tell you’re in motion? You can look at the scenery

outside, but that’s irrelevant, because we could argue that the trees

and cows are moving while you stand still. (The Martians say both

train and scenery are moving.) The real point is whether you can

detect your motion without reference to any external object. You

can hear the repetitive thunk-thunk-thunk as the train passes from

one piece of track to the next, but again this is just a reference to an

external object — all that proves is that you’re moving relative to

the tracks, but is there any way to tell that you’re moving in some

absolute sense? Assuming no interaction with the outside world, is

there any experiment you can do that will give a different result

when the train is in motion than when it’s at rest?

You could

if translation symmetry was violated. If the laws of physics were

different in different places, then as the train moved it would pass

through them. “Riding over” these regions would be like riding over

the pieces of track, but you would be able to detect the transition

from one region to the next simply because experiments inside the

train came out different, without referring to any external objects.

Rather than the thunk-thunk-thunk of the rails, you would detect

Section 2.2

The principle of inertia

39

increases and decreases in some quantity such as the gravitational

constant G, or the speed of light, or the mass of the electron.

We can therefore conclude that the following two hypotheses are

closely related.

The principle of inertia

The results of experiments don’t depend on the straight-line,

constant-speed motion of the apparatus.

Translation symmetry

The laws of physics are the same at every point in space. Specifically,

experiments don’t give different results just because you set up your

apparatus in a different place.

A state of absolute rest

example 1

Suppose that translation symmetry is violated. The laws of phys-

ics are different in one region of space than in another. Cruising in

our spaceship, we monitor the fluctuations in the laws of physics

by watching the needle on a meter that measures some funda-

mental quantity such as the gravitational constant. We make a

short blast with the ship’s engines and turn them off again. Now

we see that the needle is wavering more slowly, so evidently it’s

taking us more time to move from one region to the next. We

keep on blasting with the ship’s engines until the fluctuations stop

entirely. Now we know that we’re in a state of absolute rest. The

violation of translation symmetry logically resulted in a violation of

the principle of inertia.

self-check A

Suppose you do an experiment to see how long it takes for a rock to

drop one meter. This experiment comes out different if you do it on the

moon. Does this violate translation symmetry?

. Answer, p. 179

2.3 Momentum

Conservation of momentum

Let’s return to the impossible story of Jen Yu and Iron Arm

Lu on page 37. For simplicity, we’ll model them as two identical,

featureless pool balls, a. This may seem like a drastic simplification,

but even a collision between two human bodies is really just a series

of many collisions between atoms. The film shows a series of instants

in time, viewed from overhead.

The light-colored ball comes in,

hits the darker ball, and rebounds. It seems strange that the dark

ball has such a big effect on the light ball without experiencing

any consequences itself, but how can we show that this is really

impossible?

40

Chapter 2

Conservation of Momentum

a / How can we prove that this col-

lision is impossible?

We can show it’s impossible by looking at it in a different frame

of reference, b. This camera follows the light ball on its way in, so

in this frame the incoming light ball appears motionless. (If you

ever get hauled into court on an assault charge for hitting someone,

try this defense: “Your honor, in my fist’s frame of reference, it

was his face that assaulted my knuckles!”) After the collision, we

let the camera keep moving in the same direction, because if we

didn’t, it wouldn’t be showing us an inertial frame of reference.

To help convince yourself that figures a and b represent the same

motion seen in two different frames, note that both films agree on

the distances between the balls at each instant. After the collision,

frame b shows the light ball moving twice as fast as the dark ball;

an observer who prefers frame a explains this by saying that the

camera that produced film b was moving one way, while the ball

was moving the opposite way.

b / The collision of figure a is

viewed in a different frame of ref-

erence.

Figures a and b record the same events, so if one is impossible,

the other is too. But figure b is definitely impossible, because it

violates conservation of energy. Before the collision, the only kinetic

energy is the dark ball’s. After the collision, light ball suddenly has

some energy, but where did that energy come from? It can only

have come from the dark ball. The dark ball should then have lost

some energy, which it hasn’t, since it’s moving at the same speed as

before.

Figure c shows what really does happen. This kind of behavior

is familiar to anyone who plays pool. In a head-on collision, the

Section 2.3

Momentum

41

incoming ball stops dead, and the target ball takes all its energy

and flies away. In c/1, the light ball hits the dark ball. In c/2, the

camera is initially following the light ball; in this frame of reference,

the dark ball hits the light one (“Judge, his face hit my knuckles!”).

The frame of reference shown in c/3 is particularly interesting. Here

the camera always stays at the midpoint between the two balls. This

is called the center-of-mass frame of reference.

c / This is what really happens.

Three films represent the same

collision viewed in three differ-

ent frames of reference. Energy

is conserved in all three frames.

Figure d shows a less cumber-

some way of representing the

same thing.

self-check B

In each picture in figure c/1, mark an x at the point half-way in between

the two balls. This series of five x’s represents the motion of the camera

that was used to make the bottom film. How fast is the camera moving?

Does it represent an inertial frame of reference?

. Answer, p. 179

What’s special about the center-of-mass frame is its symmetry.

In this frame, both balls have the same initial speed. Since they

start out with the same speed, and they have the same mass, there’s

no reason for them to behave differently from each other after the

collision. By symmetry, if the light ball feels a certain effect from

the dark ball, the dark ball must feel the same effect from the light

42

Chapter 2

Conservation of Momentum

ball.

This is exactly like the rules of accounting. Let’s say two big

corporations are doing business with each other. If Glutcorp pays a

million dollars to Slushco, two things happen: Glutcorp’s bank ac-

count goes down by a million dollars, and Slushco’s rises by the same

amount. The two companies’ books have to show transactions on

the same date that are equal in size, but one is positive (a payment)

and one is negative. What if Glutcorp records −1, 000, 000 dollars,

but Slushco’s books say +920, 000? This indicates that a law has

been broken; the accountants are going to call the police and start

looking for the employee who’s driving a new 80,000-dollar Jaguar.

Money is supposed to be conserved.

In figure c, let’s define velocities as positive if the motion is

toward the top of the page. In figure c/1 let’s say the incoming light

ball’s velocity is 1 m/s.

velocity (meters per second)

before the colli-

after the collision

change

sion

d / A

less

cumbersome

rep-

0

1

+1

resentation of figure c.

The

1

0

−1

collision is shown as a graph

The books balance. The light ball’s payment, −1, matches the dark

of position, x , versus time, t.

ball’s receipt, +1. Everything also works out fine in the center of

By distorting the graph-paper

grid, we can describe the same

mass frame, c/3:

collision in the two other frames

velocity (meters per second)

of reference. Cf. figure e.

before the colli-

after the collision

change

sion

−0.5

+0.5

+1

+0.5

−0.5

−1

self-check C

Make a similar table for figure c/2. What do you notice about the change

in velocity when you compare the three tables?

. Answer, p. 179

Accounting works because money is conserved. Apparently, some-

e / A visual shorthand for de-

thing is also conserved when the balls collide. We call it momentum.

scribing the relationship between

Momentum is not the same as velocity, because conserved quanti-

two frames of reference as in

ties have to be additive. Our pool balls are like identical atoms, but

figure d: the gray rectangle rep-

atoms can be stuck together to form molecules, people, and plan-

resents the original, undistorted,

ets. Because conservation laws work by addition, two atoms stuck

graph paper, while the super-

imposed grid shows a different

together and moving at a certain velocity must have double the mo-

frame.

mentum that a single atom would have had. We therefore define

momentum as velocity multiplied by mass.

Conservation of momentum

The quantity defined by momentum = mv is conserved.

Section 2.3

Momentum

43

This is our second example of Noether’s theorem:

symmetry

conserved quantity

time symmetry

⇒

mass-energy

translation symmetry

⇒

momentum

Conservation of momentum for pool balls

example 2

. Is momentum conserved in figure c/1?

. We have to check whether the total initial momentum is the

same as the total final momentum.

dark ball’s initial momentum + light ball’s initial momentum

=?

dark ball’s final momentum + light ball’s final momentum

Yes, momentum was conserved:

0 + mv = mv + 0

Ice skaters push off from each other

example 3

If the ice skaters in figure f have equal masses, then left-right

(mirror) symmetry implies that they moved off with equal speeds

in opposite directions. Let’s check that this is consistent with con-

servation of momentum:

left skater’s initial momentum + right skater’s initial momentum

=?

f / Example 3.

left skater’s final momentum + right skater’s final momentum

Momentum was conserved:

0 + 0 = m × (−v ) + mv

This is an interesting example, because if these had been pool

balls instead of people, we would have accused them of violat-

ing conservation of energy. Initially there was zero kinetic energy,

and at the end there wasn’t zero. (Note that the energies at the

end don’t cancel, because kinetic energy is always positive, re-

gardless of direction.) The mystery is resolved because they’re

people, not pool balls. They both ate food, and they therefore

have chemical energy inside their bodies:

food energy → kinetic energy + kinetic energy + heat

44

Chapter 2

Conservation of Momentum

Unequal masses

example 4

. Suppose the skaters have unequal masses: 50 kg for the one

on the left, and 55 kg for the other. The more massive skater, on

the right, moves off at 1.0 m/s. How fast does the less massive

skater go?

. Their momenta (plural of momentum) have to be the same

amount, but with opposite signs. The less massive skater must

have a greater velocity if her momentum is going to be as much

as the more massive one’s.

0 + 0 = (50 kg)(−v ) + (55 kg)(1.0 m/s)

(50 kg)(v ) = (55 kg)(1.0 m/s)

(55 kg)

v =

(1.0 m/s)

50 kg

= 1.1 m/s

Momentum compared to kinetic energy

Momentum and kinetic energy are both measures of the amount

of motion, and a sideshow in the Newton-Leibniz controversy over

who invented calculus was an argument over which quantity was

the “true” measure of motion. The modern student can certainly

be excused for wondering why we need both quantities, when their

complementary nature was not evident to the greatest minds of the

1700’s. The following table highlights their differences.

Kinetic energy. . .

Momentum. . .

has no direction in space.

has a direction in space.

is always positive, and cannot

cancels with momentum in the

cancel out.

opposite direction.

can be traded for forms of en-

is always conserved.

ergy that do not involve mo-

tion.

KE is not a conserved

quantity by itself.

is quadrupled if the velocity is

is doubled if the velocity is dou-

doubled.

bled.

Here are some examples that show the different behaviors of the

two quantities.

A spinning coin

example 5

A spinning coin has zero total momentum, because for every

moving point, there is another point on the opposite side that can-

cels its momentum. It does, however, have kinetic energy.

Momentum and kinetic energy in firing a rifle

example 6

The rifle and bullet have zero momentum and zero kinetic energy

to start with. When the trigger is pulled, the bullet gains some mo-

g / Example 5.

mentum in the forward direction, but this is canceled by the rifle’s

Section 2.3

Momentum

45

backward momentum, so the total momentum is still zero. The

kinetic energies of the gun and bullet are both positive numbers,

however, and do not cancel. The total kinetic energy is allowed

to increase, because both objects’ kinetic energies are destined

to be dissipated as heat — the gun’s “backward” kinetic energy

does not refrigerate the shooter’s shoulder!

The wobbly earth

example 7

As the moon completes half a circle around the earth, its motion

reverses direction. This does not involve any change in kinetic

energy, because the moon doesn’t speed up or slow down, nor

is there any change in gravitational energy, because the moon

stays at the same distance from the earth.1 The reversed veloc-

ity does, however, imply a reversed momentum, so conservation

of momentum tells us that the earth must also change its mo-

mentum. In fact, the earth wobbles in a little “orbit” about a point

below its surface on the line connecting it and the moon. The two

bodies’ momenta always point in opposite directions and cancel

each other out.

The earth and moon get a divorce

example 8

Why can’t the moon suddenly decide to fly off one way and the

earth the other way? It is not forbidden by conservation of mo-

mentum, because the moon’s newly acquired momentum in one

direction could be canceled out by the change in the momentum

h / Example 8.

of the earth, supposing the earth headed the opposite direction

at the appropriate, slower speed. The catastrophe is forbidden by

conservation of energy, because both their kinetic energies would

have increased greatly.

Momentum and kinetic energy of a glacier

example 9

A cubic-kilometer glacier would have a mass of about 1012 kg —

1 followed by 12 zeroes. If it moves at a speed of 0.00001 m/s,

then its momentum would be 10, 000, 000 kg·m/s. This is the kind

of heroic-scale result we expect, perhaps the equivalent of the

space shuttle taking off, or all the cars in LA driving in the same

direction at freeway speed. Its kinetic energy, however, is only 50

joules, the equivalent of the calories contained in a poppy seed

or the energy in a drop of gasoline too small to be seen without

a microscope. The surprisingly small kinetic energy is because

kinetic energy is proportional to the square of the velocity, and the

square of a small number is an even smaller number.

Force

1Actually these statements are both only approximately true. The moon’s

orbit isn’t exactly a circle.

46

Chapter 2

Conservation of Momentum

Definition of force

When momentum is being transferred, we refer to the rate of

transfer as the force.2 The metric unit of force is the newton (N).

The relationship between force and momentum is like the relation-

ship between power and energy, or the one between your cash flow

and your bank balance:

conserved quantity

rate of transfer

name

units

name

units

energy

joules (J)

power

watts (W)

momentum

kg·m/s

force

newtons

(N)

A bullet

example 10

. A bullet emerges from a gun with a momentum of 1.0 kg·m/s,

after having been acted on for 0.01 seconds by the force of the

gases from the explosion of the gunpowder. What was the force

on the bullet?

. The force is3

1.0 = 100 newtons.

0.01

There’s no new physics happening here, just a definition of the

word “force.” Definitions are neither right nor wrong, and just be-

cause the Chinese call it

instead, that doesn’t mean they’re incor-

rect. But when Isaac Newton first started using the term “force”

according to this technical definition, people already had some def-

inite ideas about what the word meant.

In some cases Newton’s definition matches our intuition. In ex-

ample 10, we divided by a small time, and the result was a big force;

this is intuitively reasonable, since we expect the force on the bullet

to be strong.

Forces occur in equal-strength pairs

In other situations, however, our intuition rebels against reality.

Extra protein

example 11

. While riding my bike fast down a steep hill, I pass through a

cloud of gnats, and one of them goes into my mouth. Compare

my force on the gnat to the gnat’s force on me.

. Momentum is conserved, so the momentum gained by the gnat

equals the momentum lost by me. Momentum conservation holds

true at every instant over the fraction of a second that it takes for

2This definition is known as Newton’s second law of motion. Don’t memorize

that!

3This is really only an estimate of the average force over the time it takes for

the bullet to move down the barrel. The force probably starts out stronger than

this, and then gets weaker because the gases expand and cool.

Section 2.3

Momentum

47

the collision to happen. The rate of transfer of momentum out

of me must equal the rate of transfer into the gnat. Our forces

on each other have the same strength, but they’re in opposite

directions.

Most people would be willing to believe that the momentum gained

by the gnat is the same as the momentum lost by me, but they would

not believe that the forces are the same strength.

Nevertheless,

the second statement follows from the first merely as a matter of

definition. Whenever two objects, A and B, interact, A’s force on B

is the same strength as B’s force on A, and the forces are in opposite

directions.4

(A on B) = −(B on A)

Using the metaphor of money, suppose Alice and Bob are adrift in

a life raft, and pass the time by playing poker. Money is conserved,

so if they count all the money in the boat every night, they should

always come up with the same total. A completely equivalent state-

ment is that their cash flows are equal and opposite. If Alice is

winning five dollars per hour, then Bob must be losing at the same

rate.

This statement about equal forces in opposite directions implies

to many students a kind of mystical principle of equilibrium that

explains why things don’t move. That would be a useless principle,

since it would be violated every time something moved.5 The ice

skaters of figure f on page 44 make forces on each other, and their

forces are equal in strength and opposite in direction. That doesn’t

mean they won’t move. They’ll both move — in opposite directions.

The fallacy comes from trying to add things that it doesn’t make

sense to add, as suggested by the cartoon in figure i. We only add

forces that are acting on the same object. It doesn’t make sense to

say that the skaters’ forces on each other add up to zero, because

i / It

doesn’t

make

sense

to

it doesn’t make sense to add them. One is a force on the left-hand

add his debts to her assets.

skater, and the other is a force on the right-hand skater.

In figure j, my fingers’ force and my thumbs’ force are both acting

on the bathroom scale. It does make sense to add these forces, and

they may possibly add up to zero, but that’s not guaranteed by the

laws of physics. If I throw the scale at you, my thumbs’ force is

stronger that my fingers’, and the forces no longer cancel:

(fingers on scale) 6= −(thumbs on scale).

j / I

squeeze

the

bathroom

scale. It does make sense to add

What’s guaranteed by conservation of momentum is a whole differ-

my fingers’ force to my thumbs’,

because they both act on the

4This is called Newton’s third law. Don’t memorize that name!

same object — the scale.

5During the Scopes monkey trial, William Jennings Bryan claimed that every

time he picked his foot up off the ground, he was violating the law of gravity.

48

Chapter 2

Conservation of Momentum

ent relationship:

(fingers on scale) = −(scale on fingers)

(thumbs on scale) = −(scale on thumbs)

The force of gravity

How much force does gravity make on an object? From everyday

experience, we know that this force is proportional to the object’s

mass.6 Let’s find the force on a one-kilogram object. If we release

this object from rest, then after it has fallen one meter, its kinetic

energy equals the strength of the gravitational field,

10 joules per kilogram per meter × 1 kilogram × 1 meter = 10 joules.

Using the equation for kinetic energy and doing a little simple alge-

bra, we find that its final velocity is 4.4 m/s. It starts from 0 m/s,

and ends at 4.4 m/s, so its average velocity is 2.2 m/s, and the time

takes to fall one meter is therefore (1 m)/(2.2 m/s)=0.44 seconds.

Its final momentum is 4.4 units, so the force on it was evidently

4.4 = 10 newtons.

0.44

This is like one of those card tricks where the magician makes you go

through a bunch of steps so that you end up revealing the card you

had chosen — the result is just equal to the gravitational field, 10,

but in units of newtons! If algebra makes you feel warm and fuzzy,

you may want to replay the derivation using symbols and convince

yourself that it had to come out that way. If not, then I hope the

numerical result is enough to convince you of the general fact that

the force of gravity on a one-kilogram mass equals g. For masses

other than one kilogram, we have the handy-dandy result that

(force of gravity on a mass m) = mg.

In other words, g can be interpreted not just as the gravitational

energy per kilogram per meter of height, but also as the gravitational

force per kilogram.

Motion in two dimensions

Projectile motion

Galileo was an innovator in more than one way. He was arguably

the inventor of open-source software: he invented a mechanical cal-

culating device for certain engineering applications, and rather than

keeping the device’s design secret as his competitors did, he made

it public, but charged students for lessons in how to use it. Not

6This follows from the additivity of forces.

Section 2.3

Momentum

49

only that, but he was the first physicist to make money as a mili-

tary consultant. Galileo understood projectiles better than anyone

else, because he understood the principle of inertia. Even if you’re

not planning on a career involving artillery, projectile motion is a

good thing to learn about because it’s an example of how to handle

motion in two or three dimensions.

Figure k shows a ball in the process of falling — or rising, it

really doesn’t matter which. Let’s say the ball has a mass of one

kilogram, each square in the grid is 10 meters on a side, and the

positions of the ball are shown at time intervals of one second. The

earth’s gravitational force on the ball is 10 newtons, so with each

second, the ball’s momentum increases by 10 units, and its speed

also increases by 10 m/s. The ball falls 10 m in the first second, 20

m in the next second, and so on.

self-check D

What would happen if the ball’s mass was 2 kilograms?

. Answer, p.

k / A ball is falling (or rising).

179

Now let’s look at the ball’s motion in a new frame of reference,

l, which is moving at 10 meters per second to the left compared to

the frame of reference used in figure k. An observer in this frame

of reference sees the ball as moving to the right by 10 meters every

second. The ball traces an arc of a specific mathematical type called

a parabola:

1 step over and 1 step down

1 step over and 2 steps down

1 step over and 3 steps down

1 step over and 4 steps down

. . .

It doesn’t matter which frame of reference is the “real” one. Both

diagrams show the possible motion of a projectile. The interesting

point here is that the vertical force of gravity has no effect on the

l / The same ball is viewed in

horizontal motion, and the horizontal motion also has no effect on

a frame of reference that is

what happens in the vertical motion. The two are completely in-

moving horizontally.

dependent. If the sun is directly overhead, the motion of the ball’s

shadow on the ground seems perfectly natural: there are no horizon-

tal forces, so it either sits still or moves at constant velocity. (Zero

force means zero rate of transfer of momentum.) The same is true

if we shine a light from one side and cast the ball’s shadow on the

wall. Both shadows obey the laws of physics.

The moon

example 12

In example 12 on page 27, I promised an explanation of how New-

ton knew that the gravitational field experienced by the moon due

to the earth was 1/3600 of the one we feel here on the earth’s

m / The drops of water travel

surface. The radius of the moon’s orbit had been known since

in parabolic arcs.

ancient times, so Newton knew its speed to be 1,100 m/s (ex-

50

Chapter 2

Conservation of Momentum

pressed in modern units). If the earth’s gravity wasn’t acting on

the moon, the moon would fly off straight, along the straight line

shown in figure o, and it would cover 1,100 meters in one second.

We observe instead that it travels the arc of a circle centered on

the earth. Straightforward geometry shows that the amount by

o / Example 12.

which the arc drops below the straight line is 1.6 millimeters. Near

the surface of the earth, an object falls 5 meters in one second,7

which is indeed about 3600 times greater than 1.6 millimeters.

The tricky part about this argument is that although I said the

path of a projectile was a parabola, in this example it’s a circle.

What’s going on here? What’s different here is that as the moon

moves 1,100 meters, it changes its position relative to the earth,

so down is now in a new direction. We’ll discuss circular motion

more carefully soon, but in this example, it really doesn’t matter.

The curvature of the arc is so gentle that a parabola and a circle

would appear almost identical. (Actually the curvature is so gentle

— 1.6 millimeters over a distance of 1,100 meters! — that if I had

drawn the figure to scale, you wouldn’t have even been able to tell

that it wasn’t straight.)

As an interesting historical note, Newton claimed that he first

did this calculation while confined to his family’s farm during the

plague of 1666, and found the results to “answer pretty nearly.”

His notebooks, however, show that although he did the calcula-

tion on that date, the result didn’t quite come out quite right, and

he became uncertain about whether his theory of gravity was cor-

rect as it stood or needed to be modified. Not until 1675 did he

learn of more accurate astronomical data, which convinced him

that his theory didn’t need to be tinkered with. It appears that he

rewrote his own life story a little bit in order to make it appear that

his work was more advanced at an earlier date, which would have

helped him in his dispute with Leibniz over priority in the invention

of calculus.

The memory of motion

There’s another useful way of thinking about motion along a

curve. In the absence of a force, an object will continue moving

in the same speed and in the same direction. One of my students

invented a wonderful phrase for this: the memory of motion. Over

the first second of its motion, the ball in figure n moved 1 square over

and 1 square down, which is 10 meters and 10 meters. The default

for the next one-second interval would be to repeat this, ending up

n / The

memory

of

motion:

at the location marked with the first dashed circle. The earth’s 10-

the default would be for the ball

to continue doing what it was

newton gravitational force on the ball, however, changes the vertical

already doing. The force of grav-

part of the ball’s momentum by 10 units. The ball actually ends up

ity makes it deviate downward,

ending up one square below the

7Its initial speed is 0, and its final speed is 10 m/s, so its average speed is 5

default.

m/s over the first second of falling.

Section 2.3

Momentum

51

10 meters (1 square) below the default.

Circular motion

Figure q shows how to apply the memory-of-motion idea to cir-

cular motion. It should convince you that only an inward force is

needed to produce circular motion. One of the reasons Newton was

the first to make any progress in analyzing the motion of the planets

around the sun was that his contemporaries were confused on this

point. Most of them thought that in addition to an attraction from

the sun, a second, forward force must exist on the planets, to keep

them from slowing down. This is incorrect Aristotelian thinking;

objects don’t naturally slow down. Car 1 in figure p only needs a

forward force in order to cancel out the backward force of friction;

the total force on it is zero. Similarly, the forward and backward

forces on car 2 are canceling out, and the only force left over is the

inward one. There’s no friction in the vacuum of outer space, so if

car 2 was a planet, the backward force wouldn’t exist; the forward

force wouldn’t exist either, because the only force would be the force

of the sun’s gravity.

p / The forces on car 1 can-

cel, and the total force on it is

q / A large number of gentle taps gives a good approximation to

zero. The forward and backward

circular motion. A steady inward force would give exactly circular motion.

forces on car 2 also cancel. Only

the inward force remains.

One confusing thing about circular motion is that it often tempts

us psychologically to adopt a noninertial frame of reference. Figure

r shows a bowling ball in the back of a turning pickup truck. Each

panel gives a view of the same events from a different frame of

reference. The frame of reference r/1, attached to the turning truck,

is noninertial, because it changes the direction of its motion. The

ball violates conservation of energy by accelerating from rest for

no apparent reason. Is there some mysterious outward force that is

slamming the ball into the side of the truck’s bed? No. By analyzing

everything in a proper inertial frame of reference, r/2, we see that

it’s the truck that swerves and hits the ball. That makes sense,

52

Chapter 2

Conservation of Momentum

because the truck is interacting with the asphalt.

t / Tycho Brahe made his name

as an astronomer by showing that

r / A bowling ball is in the back of a pickup truck turning left.

The

the bright new star, today called

motion is viewed first in a frame that turns along with the truck, 1, and

a supernova, that appeared in

then in an inertial frame, 2.

the skies in 1572 was far beyond

the Earth’s atmosphere.

This,

along with Galileo’s discovery of

sunspots, showed that contrary

2.4 Newton’s triumph

to Aristotle, the heavens were

Isaac Newton’s greatest triumph was his explanation of the motion

not

perfect

and

unchanging.

of the planets in terms of universal physical laws. It was a tremen-

Brahe’s fame as an astronomer

brought him patronage from King

dous psychological revolution: for the first time, both heaven and

Frederick II, allowing him to carry

earth were seen as operating automatically according to the same

out his historic high-precision

rules.

measurements of the planets’

Newton wouldn’t have been able to figure out why the planets

motions. A contradictory charac-

ter, Brahe enjoyed lecturing other

move the way they do if it hadn’t been for the astronomer Tycho

nobles about the evils of dueling,

Brahe (1546-1601) and his protege Johannes Kepler (1571-1630),

but had lost his own nose in a

who together came up with the first simple and accurate description

youthful duel and had it replaced

of how the planets actually do move. The difficulty of their task is

with a prosthesis made of an

suggested by figure s, which shows how the relatively simple orbital

alloy of gold and silver. Willing to

motions of the earth and Mars combine so that as seen from earth

endure scandal in order to marry

Mars appears to be staggering in loops like a drunken sailor.

a peasant, he nevertheless used

the feudal powers given to him by

the king to impose harsh forced

labor on the inhabitants of his

Brahe, the last of the great naked-eye astronomers, collected ex-

parishes. The result of their work,

tensive data on the motions of the planets over a period of many

an Italian-style palace with an

years, taking the giant step from the previous observations’ accuracy

observatory on top, surely ranks

of about 10 minutes of arc (10/60 of a degree) to an unprecedented

as one of the most luxurious

1 minute. The quality of his work is all the more remarkable consid-

science labs ever built. He died

ering that his observatory consisted of four giant brass protractors

of a ruptured bladder after falling

from a wagon on the way home

mounted upright in his castle in Denmark. Four different observers

from a party — in those days, it

would simultaneously measure the position of a planet in order to

was considered rude to leave the

check for mistakes and reduce random errors.

dinner table to relieve oneself.

With Brahe’s death, it fell to his former assistant Kepler to try

Section 2.4

Newton’s triumph

53

s / As the Earth and Mars revolve around the sun at different rates,

the combined effect of their motions makes Mars appear to trace a

strange, looped path across the background of the distant stars.

to make some sense out of the volumes of data. Kepler, in con-

tradiction to his late boss, had formed a prejudice, a correct one

as it turned out, in favor of the theory that the earth and planets

revolved around the sun, rather than the earth staying fixed and

everything rotating about it. Although motion is relative, it is not

just a matter of opinion what circles what. The earth’s rotation

and revolution about the sun make it a noninertial reference frame,

which causes detectable violations of Newton’s laws when one at-

tempts to describe sufficiently precise experiments in the earth-fixed

frame. Although such direct experiments were not carried out until

the 19th century, what convinced everyone of the sun-centered sys-

tem in the 17th century was that Kepler was able to come up with

a surprisingly simple set of mathematical and geometrical rules for

describing the planets’ motion using the sun-centered assumption.

After 900 pages of calculations and many false starts and dead-end

ideas, Kepler finally synthesized the data into the following three

laws:

Kepler’s elliptical orbit law

The planets orbit the sun in elliptical orbits with the sun at

one focus.

Kepler’s equal-area law

The line connecting a planet to the sun sweeps out equal areas

in equal amounts of time.

54

Chapter 2

Conservation of Momentum

Kepler’s law of periods

Let T , called the planet’s period, be the time required for a

planet to orbit the sun, and let a be the long axis of the ellipse.

Then T 2 is proportional to a3.

u / An ellipse is a circle that

Although the planets’ orbits are ellipses rather than circles, most

has been distorted by shrinking

are very close to being circular. The earth’s orbit, for instance, is

and stretching along perpendicu-

only flattened by 1.7% relative to a circle. In the special case of a

lar axes.

planet in a circular orbit, the two foci (plural of “focus”) coincide

at the center of the circle, and Kepler’s elliptical orbit law thus says

that the circle is centered on the sun. The equal-area law implies

that a planet in a circular orbit moves around the sun with constant

speed. For a circular orbit, the law of periods then amounts to a

statement that T 2 is proportional to r3, where r is the radius. If

all the planets were moving in their orbits at the same speed, then

the time for one orbit would only increase with the circumference

of the circle, so we would have a simple proportionality between T

and r. Since this is not the case, we can interpret the law of periods

to mean that different planets orbit the sun at different speeds. In

fact, the outer planets move more slowly than the inner ones.

v / An ellipse can be constructed

Jupiter and Uranus

example 13

by tying a string to two pins and

. The planets Jupiter and Uranus have very nearly circular orbits,

drawing like this with the pencil

and the radius of Uranus’s orbit is about four times grater than

stretching the string taut.

Each

that of Jupiter’s orbit. Compare their orbital periods.

pin constitutes one focus of the

ellipse.

. If all the planets moved at the same speed, then it would take

Uranus four times longer to complete the four-times-greater cir-

cumference of its orbit. However, the law of periods tells us that

this isn’t the case. We expect Uranus to take more than four times

as long to orbit the sun.

The law of periods is stated as a proportionality, and proportional-

ities are statements about quantities in proportion to one another,

i.e.. about division. We’re given information about Uranus’s or-

bital radius divided by Jupiter’s, and what we should expect to get

out is information about Uranus’s period divided by Jupiter’s. Let’s

w / If

the

time

interval

taken

call the latter ratio y . Then we’re looking for a number y such that

by the planet to move from P to Q

is equal to the time interval from

y 2 = 43,

R to S, then according to Kepler’s

equal-area law, the two shaded

i.e.,

areas are equal.

The planet

is moving faster during interval

RS than it did during PQ, which

y × y = 4 × 4 × 4

Newton later determined was due

y × y = 64

to the sun’s gravitational force

accelerating it.

The equal-area

y = 8

law predicts exactly how much it

will speed up.

The law of periods predicts that Uranus’s period will be eight times

greater than Jupiter’s, which is indeed what is observed (to within

the precision to be expected since the given figure of 4 was just

Section 2.4

Newton’s triumph

55

stated roughly as a whole number, for convenience in calcula-

tion).

What Newton discovered was the reasons why Kepler’s laws were

true: he showed that they followed from his laws of motion. From a

modern point of view, conservation laws are more fundamental than

Newton’s laws, so rather than following Newton’s approach, it makes

more sense to look for the reasons why Kepler’s laws follow from

conservation laws. The equal-area law is most easily understood as

a consequence of conservation of angular momentum, which is a new

conserved quantity to be discussed in chapter 3. The proof of the

elliptical orbit law is a little too mathematical to be appropriate for

this book, but the interested reader can find the proof in chapter 15

of my online book Light and Matter.

x / Connecting Kepler’s law of pe-

riods to the laws of physics.

The law of periods follows directly from the physics we’ve already

covered. Consider the example of Jupiter and Uranus. We want to

show that the result of example 13 is the only one that’s consistent

with conservation of energy and momentum, and Newton’s law of

gravity. Since Uranus takes eight times longer to cover four times

the distance, it’s evidently moving at half Jupiter’s speed. In figure

x, the distance Jupiter covers from A to B is therefore twice the

distance Uranus covers, over the same time, from D to E. If there

hadn’t been any gravitational force from the sun, Jupiter would

have ended up at C, and Uranus at F. The distance from B to C is

a measure of how much force acted on Jupiter, and likewise for the

very small distance from E to F. We find that BC is 16 millimeters

on this scale drawing, and EF is 1 mm, but this is exactly what

we expect from Newton’s law of gravity: quadrupling the distance

should give 1/16 the force.

56

Chapter 2

Conservation of Momentum

2.5 Work

Imagine a black box8, containing a gasoline-powered engine, which

is designed to reel in a steel cable of length d, exerting a certain

force F .

If we use this black box was to lift a weight, then by the time it

has pulled in its whole cable, it will have lifted the weight through

a height d. The force F is barely capable of lifting a weight m if

y / The black box does work

F = mg, and if it does this, then the upward force from the cable

by reeling in its cable.

exactly cancels the downward force of gravity, so the weight will

rise at constant speed, without changing its kinetic energy. Only

gravitational energy is transferred into the weight, and the amount

of gravitational energy is mgd, which equals F d. By conservation

of energy, this must also be the amount of energy lost from the

chemical energy of the gasoline inside the box.9

Now what if we use the black box to pull a plow? The energy in-

crease in the outside world is of a different type than before: mainly

heat created by friction between the dirt and the ploughshare. The

box, however, only communicates with the outside world via the

hole through which its cable passes. The amount of chemical en-

ergy lost by the gasoline can therefore only depend on F and d, so

again the amount of energy transferred must equal F d.

The same reasoning can in fact be applied no matter what the

cable is being used to do. There must always be a transfer of energy

from the box to the outside world that is equal to F d. In general,

when energy is transferred, we refer to the amount of energy trans-

ferred as work, W . If, as in the example of the black box, the motion

of the object to which the force is applied is in the same direction

as the force, then W = F d.

z / The baseball pitcher put ki-

netic energy into the ball, so he

did work on it. To do the greatest

possible amount of work, he ap-

plied the greatest possible force

over the greatest possible dis-

tance.

If the motion is in the opposite direction compared to the force,

then W = −F d; the negative work is to be interpreted as energy

removed from the object to which the force was applied. For ex-

8“Black box” is a traditional engineering term for a device whose inner work-

ings we don’t care about.

9For conceptual simplicity, we ignore the transfer of heat energy to the outside

world via the exhaust and radiator. In reality, the sum of these energies plus the

useful kinetic energy transferred would equal W .

Section 2.5

Work

57

ample, if Superman gets in front of an oncoming freight train, and

brings it to a stop, he’s decreased its energy rather than increasing

it. In a normal gasoline-powered car, stepping on the brakes takes

away the car’s kinetic energy (doing negative work on it), and turns

it into heat in the brake shoes. In an electric or hybrid-electric car,

the car’s kinetic energy is transformed back into electrical energy to

be used again.

58

Chapter 2

Conservation of Momentum

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

The beer bottle shown in the figure is resting on a table in the

dining car of a train. The tracks are straight and level. What can

you tell about the motion of the train? Can you tell whether the

train is currently moving forward, moving backward, or standing

still? Can you tell what the train’s speed is?

2

You’re a passenger in the open basket hanging under a hot-

air balloon. The balloon is being carried along by the wind at a

constant velocity. If you’re holding a flag in your hand, will the flag

wave? If so, which way? (Based on a question from PSSC Physics.)

3

Driving along in your car, you take your foot off the gas, and

Problem 1.

your speedometer shows a reduction in speed. Describe an inertial

frame in which your car was speeding up during that same period of

time.

4

If all the air molecules in the room settled down in a thin

film on the floor, would that violate conservation of momentum?

Conservation of energy?

5

A bullet flies through the air, passes through a paperback

book, and then continues to fly through the air beyond the book.

When is there a force? When is there energy?

6

(a) Continue figure l farther to the left, and do the same for

the numerical table in the text.

Problem 2.

(b) Sketch a smooth curve (a parabola) through all the points on

the figure, including all the ones from the original figure and all the

ones you added. Identify the very top of its arc.

(c) Now consider figure k. Is the highest point shown in the figure

the top of the ball’s up-down path? Explain by comparing with your

results from parts a and b.

7

Criticize the following statement about the top panel of figure

c on page 42: In the first few pictures, the light ball is moving up

and to the right, while the dark ball moves directly to the right.

8

Figure ac on page 60 shows a ball dropping to the surface of

the earth. Energy is conserved: over the whole course of the film,

the gravitational energy between the ball and the earth decreases

by 1 joule, while the ball’s kinetic energy increases by 1 joule.

(a) How can you tell directly from the figure that the ball’s speed

isn’t staying the same?

(b) Draw what the film would look like if the camera was following

the ball.

Problems

59

(c) Explain how you can tell that in this new frame of reference,

energy is not conserved.

(d) Does this violate the strong principle of inertia? Isn’t every

frame of reference supposed to be equally valid?

Problem 8.

9

Two cars with different masses each have the same kinetic

energy. (a) If both cars have the same brakes, capable of supplying

the same force, how will the stopping distances compare? Explain.

(b) Compare the times required for the cars to stop.

10

In each of the following situations, is the work being done

positive, negative, or zero? (a) a bull paws the ground; (b) a fishing

boat pulls a net through the water behind it; (c) the water resists

the motion of the net through it; (d) you stand behind a pickup

truck and lower a bale of hay from the truck’s bed to the ground.

Explain. [Based on a problem by Serway and Faughn.]

11

Weiping lifts a rock with a weight of 1.0 N through a height

of 1.0 m, and then lowers it back down to the starting point. Bubba

pushes a table 1.0 m across the floor at constant speed, requiring

a force of 1.0 N, and then pushes it back to where it started. (a)

Compare the total work done by Weiping and Bubba. (b) Check

A bull paws the ground,

as

that your answers to part a make sense, using the definition of work:

in problem 10.

work is the transfer of energy. In your answer, you’ll need to discuss

what specific type of energy is involved in each case.

60

Chapter 2

Conservation of Momentum

A tornado touches down in Spring Hill, Kansas, May 20, 1957.

Chapter 3

Conservation of Angular

Momentum

3.1 Angular momentum

“Sure, and maybe the sun won’t come up tomorrow.” Of course,

the sun only appears to go up and down because the earth spins,

so the cliche should really refer to the unlikelihood of the earth’s

stopping its rotation abruptly during the night. Why can’t it stop?

It wouldn’t violate conservation of momentum, because the earth’s

rotation doesn’t add anything to its momentum. While California

spins in one direction, some equally massive part of India goes the

opposite way, canceling its momentum. A halt to Earth’s rotation

would entail a drop in kinetic energy, but that energy could simply

be converted into some other form, such as heat.

Other examples along these lines are not hard to find. A hydro-

61

gen atom spins at the same rate for billions of years. A high-diver

who is rotating when he comes off the board does not need to make

any physical effort to continue rotating, and indeed would be unable

to stop rotating before he hit the water.

These observations have the hallmarks of a conservation law,

but what numerical measure of rotational motion is conserved? Car

engines and old-fashioned LP records have speeds of rotation mea-

sured in rotations per minute (r.p.m.), but the number of rotations

per minute (or per second) is not a conserved quantity. For example,

the twirling figure skater in figure a can pull her arms in to increase

her r.p.m.’s.

The example of the figure skater suggests that this conserved

quantity depends on distance from the axis of rotation. We’ll notate

this distance as r, since, for an object moving in a circle around an

axis of rotation, its distance from the axis equals the radius of the

circle.

Once we realize that r is a variable that matters, it becomes clear

that the examples we’ve been considering were all examples that

would be fairly complicated mathematically, because different parts

of these objects’ masses have different values of r. For example,

the figure skater’s front teeth are farther from the axis than her

a / A

figure

skater

pulls

in

back teeth. That suggests that instead of objects with complicated

her arms so that she can execute

a spin more rapidly.

shapes, we should consider the simplest possible example, which is

a single particle, of mass m, traveling in a circle of radius r at speed

v. Experiments show that the conserved quantity in this situation

is

±mvr.

We call this quantity angular momentum. The symbol ± indicates

that angular momentum has a positive or negative sign to represent

the direction of rotation; for example, in a given problem, we could

choose to represent clockwise angular momenta as positive numbers,

and counterclockwise ones as negative. In this equation, the only

velocity that matters is velocity that is perpendicular to the radius

line; motion parallel to the radius line, i.e., directly in our out, is

neither clockwise nor counterclockwise.

A figure skater pulls her arms in

example 1

When the skater in figure a pulls her arms in, she is decreas-

ing r for all the atoms in her arms. It would violate conserva-

tion of angular momentum if she then continued rotating at the

same speed, i.e., taking the same amount of time for each revo-

lution, because her arms would be closer to the axis of rotation

and therefore have a smaller r (as well as a smaller v because

they would be completing a smaller circle in the same time). This

is impossible because it would violate conservation of angular

momentum. If her total angular momentum is to remain constant,

the decrease in angular momentum for her arms must be com-

62

Chapter 3

Conservation of Angular Momentum

pensated for by an overall increase in her rate of rotation. That

is, by pulling her arms in, she substantially reduces the time for

each rotation.

b / Example 2: An early photo-

graph of an old-fashioned long-

jump.

A longjump

example 2

In figure b, the jumper wants to get his feet out in front of him so

he can keep from doing a “face plant” when he lands. Bringing his

feet forward would involve a certain quantity of counterclockwise

rotation, but he didn’t start out with any rotation when he left the

ground. Suppose we consider counterclockwise as positive and

clockwise as negative. The only way his legs can acquire some

positive rotation is if some other part of his body picks up an equal

amount of negative rotation. This is why he swings his arms up

behind him, clockwise.

c / Example 3.

Changing the axis

example 3

An object’s angular momentum can be different depending on the

axis about which it rotates, because r is defined relative to the

axis. Figure c shows two double-exposure photographs a viola

player tipping the bow in order to cross from one string to another.

Much more angular momentum is required when playing near the

bow’s handle, called the frog, as in the panel on the right; not

Section 3.1

Angular momentum

63

only are most of the atoms in the bow are at greater distances, r ,

from the axis of rotation, but the ones in the tip also have more

velocity, v . It is difficult for the player to quickly transfer a large

angular momentum into the bow, and then transfer it back out just

as quickly. This is one of the reasons that string players tend to

stay near the middle of the bow as much as possible.

Kepler’s equal-area law

example 4

The hypothetical planet in figure d has an orbit in which its closest

approach to the sun is at half the distance compared to the point

at which it recedes the farthest. Since angular momentum, mv r ,

is conserved, and the planet’s mass is constant, the quantity v r

must be the same at both ends of the orbit. Doubling r therefore

requires cutting v in half. If the time interval from A to B is the

same as that from C to D, then the distance from C to D must

be half as much. But this is exactly what Kepler’s equal area law

requires, since the triangular pie wedge on top needs to have half

the width to compensate for its doubled height. In other words,

the equal area law is a direct consequence of conservation of

angular momentum.

d / Example 4.

Discussion question

A

Conservation of plain old momentum, p, can be thought of as the

greatly expanded and modified descendant of Galileo’s original principle

of inertia, that no force is required to keep an object in motion. The princi-

ple of inertia is counterintuitive, and there are many situations in which it

appears superficially that a force is needed to maintain motion, as main-

tained by Aristotle. Think of a situation in which conservation of angular

momentum, L, also seems to be violated, making it seem incorrectly that

something external must act on a closed system to keep its angular mo-

mentum from “running down.”

B

The figure is a strobe photo of a pendulum bob, taken from under-

neath the pendulum looking straight up. The black string can’t be seen

in the photograph. The bob was given a slight sideways push when it

was released, so it did not swing in a plane. The bright spot marks the

center, i.e., the position the bob would have if it hung straight down at us.

Does the bob’s angular momentum appear to remain constant if we con-

sider the center to be the axis of rotation? What if we choose a different

axis?

3.2 Torque

Force is the rate of transfer of momentum. The equivalent in the

case of angular momentum is called torque (rhymes with “fork”):

amount of angular momentum transferred

torque =

time taken to transfer it

e / The

boy

makes

a

torque

on the tetherball.

Where force tells us how hard we are pushing or pulling on some-

thing, torque indicates how hard we are twisting on it.

64

Chapter 3

Conservation of Angular Momentum

Discussion question B.

Have you ever had the experience of trying to open a door by

pushing on the wrong side, the side near the hinge? It’s difficult to

do, which apparently indicates that a given amount of force produces

less torque when it’s applied close to the axis of rotation. To try

to pin down this relationship more precisely, let’s imagine hitting a

tetherball, e. The boy applies a force F to the ball for a short time

t, accelerating the ball from rest to a velocity v. Since force is the

rate of transfer of momentum, we have

mv

F =

,

t

and multiplying both sides by r gives

mvr

F r =

.

t

But ±mvr is simply the amount of angular momentum he’s given

the ball, so ±mvr/t also equals the amount of torque he applied.

The result of this example is

torque = ±F r,

where the plus or minus sign indicates whether torque would tend to

create clockwise or counterclockwise motion. This equation applies

more generally, with the caveat that F should only include the part

of the force perpendicular to the radius line.

self-check A

There are four equations on this page. Which ones are important, and

likely to be useful later?

. Answer, p. 179

To summarize, we’ve learned three conserved quantity, each of

which has a rate of transfer:

conserved quantity

rate of transfer

name

units

name

units

energy

joules (J)

power

watts (W)

momentum

kg·m/s

force

newtons

(N)

angular mo-

kg·m2/s

torque

newton-

mentum

meters

(N·m)

Section 3.2

Torque

65

Torque distinguished from force

Of course a force is necessary in order to create a torque — you

can’t twist a screw without pushing on the wrench — but force and

torque are two different things. One distinction between them is

direction. We use positive and negative signs to represent forces in

the two possible directions along a line. The direction of a torque,

however, is clockwise or counterclockwise, not a linear direction.

The other difference between torque and force is a matter of

leverage. A given force applied at a door’s knob will change the

door’s angular momentum twice as rapidly as the same force applied

f / The

plane’s

four

engines

halfway between the knob and the hinge. The same amount of force

produce zero total torque but not

produces different amounts of torque in these two cases.

zero total force.

It is possible to have a zero total torque with a nonzero total

force. An airplane with four jet engines, f, would be designed so

that their forces are balanced on the left and right. Their forces are

all in the same direction, but the clockwise torques of two of the

engines are canceled by the counterclockwise torques of the other

two, giving zero total torque.

Conversely, we can have zero total force and nonzero total torque.

A merry-go-round’s engine needs to supply a nonzero torque on it

to bring it up to speed, but there is zero total force on it. If there

was not zero total force on it, its center of mass would accelerate!

A lever

example 5

Figure g shows an example of a lever within your arm. Different

muscles are used to flex and extend the arm, because muscles

work only by contraction. The biceps flexes it.

There are three forces acting on the forearm: the force from the

biceps, the force at the elbow joint, and the force from the load

being lifted. Because the elbow joint is motionless, it is natural

to define our torques using the joint as the axis. The situation

now becomes quite simple, because the upper arm bone’s force

exerted at the elbow has r = 0, and therefore creates no torque.

We can ignore it completely. In general, we would call this the

fulcrum of the lever.

If we restrict ourselves to the case in which the forearm rotates

g / Example

5:

the

biceps

with constant angular momentum, then we know that the total

muscle flexes the arm.

torque on the forearm is zero, so the torques from the muscle and

the load must be opposite in sign and equal in absolute value:

rmuscleFmuscle = rload Fload ,

where rmuscle, the distance from the elbow joint to the biceps’

point of insertion on the forearm, is only a few cm, while rload

might be 30 cm or so. The force exerted by the muscle must

therefore be about ten times the force exerted by the load. We

thus see that this lever is a force reducer. In general, a lever may

66

Chapter 3

Conservation of Angular Momentum

be used either to increase or to reduce a force.

Why did our arms evolve so as to reduce force? In general, your

body is built for compactness and maximum speed of motion

rather than maximum force. This is the main anatomical differ-

ence between us and the Neanderthals (their brains covered the

same range of sizes as those of modern humans), and it seems

to have worked for us.

As with all machines, the lever is incapable of changing the amount

of mechanical work we can do. A lever that increases force will al-

ways reduce motion, and vice versa, leaving the amount of work

unchanged.

Discussion questions

A

You whirl a rock over your head on the end of a string, and gradually

pull in the string, eventually cutting the radius in half. What happens to

the rock’s angular momentum? What changes occur in its speed, the time

required for one revolution, and its acceleration? Why might the string

break?

B

A helicopter has, in addition to the huge fan blades on top, a smaller

propeller mounted on the tail that rotates in a vertical plane. Why?

C

The photo shows an amusement park ride whose two cars rotate in

opposite directions. Why is this a good design?

3.3 Noether’s theorem for angular momentum

Suppose a sunless planet is sitting all by itself in interstellar space,

not rotating. Then, one day, it decides to start spinning. This

doesn’t necessarily violate conservation of energy, because it could

have energy stored up, e.g., the heat in a molten core, which could be

converted into kinetic energy. It does violate conservation of angular

momentum, but even if we didn’t already know about that law of

Discussion question C.

physics, the story would seem odd. How would it decide which axis

to spin around? If it was to spontaneously start spinning about some

axis, then that axis would have to be a special, preferred direction

in space. That is, space itself would have to have some asymmetry

to it.

In reality, as I’ve already mentioned on page 15, experiments

show to a very high degree of precision that the laws of physics

are completely symmetric with respect to different directions. The

story of the planet that abruptly starts spinning is an example of

Noether’s theorem, applied to angular momentum. We now have

three such examples:

symmetry

conserved quantity

time symmetry

⇒

mass-energy

translation symmetry

⇒

momentum

rotational symmetry

⇒

angular momentum

Section 3.3

Noether’s theorem for angular momentum

67

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

You are trying to loosen a stuck bolt on your RV using a big

wrench that is 50 cm long. If you hang from the wrench, and your

mass is 55 kg, what is the maximum torque you can exert on the

√

bolt?

2

A physical therapist wants her patient to rehabilitate his in-

jured elbow by laying his arm flat on a table, and then lifting a 2.1

kg mass by bending his elbow. In this situation, the weight is 33

cm from his elbow. He calls her back, complaining that it hurts him

to grasp the weight. He asks if he can strap a bigger weight onto

his arm, only 17 cm from his elbow. How much mass should she

tell him to use so that he will be exerting the same torque? (He is

√

raising his forearm itself, as well as the weight.)

3

An object is observed to have constant angular momentum.

Can you conclude that no torques are acting on it? Explain. [Based

on a problem by Serway and Faughn.]

4

The figure shows scale drawing of a pair of pliers being

used to crack a nut, with an appropriately reduced centimeter grid.

Warning: do not attempt this at home; it is bad manners. If the

force required to crack the nut is 300 N, estimate the force required

of the person’s hand.

. Solution, p. 180

5

Two horizontal tree branches on the same tree have equal

diameters, but one branch is twice as long as the other. Give a

quantitative comparison of the torques where the branches join the

Problem 4.

trunk. [Thanks to Bong Kang.]

6

(a) Alice says Cathy’s body has zero momentum, but Bob

says Cathy’s momentum is nonzero. Nobody is lying or making a

mistake. How is this possible? Give a concrete example.

(b) Alice and Bob agree that Dong’s body has nonzero momentum,

but disagree about Dong’s angular momentum, which Alice says is

zero, and Bob says is nonzero. Explain.

7

A person of weight W stands on the ball of one foot. Find

the tension in the calf muscle and the force exerted by the shinbones

on the bones of the foot, in terms of W , a, and b. (The tension is a

measure of how tight the calf muscle has been pulled; it has units

of newtons, and equals the amount of force applied by the muscle

where it attaches to the heel.) For simplicity, assume that all the

forces are at 90-degree angles to the foot. Suggestion: Write down

Problem 7.

an equation that says the total force on the foot is zero, and another

equation saying that the total torque on the foot is zero; solve the

68

Chapter 3

Conservation of Angular Momentum

√

two equations for the two unknowns.

Problems

69

70

Chapter 3

Conservation of Angular Momentum

The clock took up two seats, and two tickets were bought for it under the

name of “Mr. Clock.

Chapter 4

Relativity

4.1 Relativity according to Einstein

Time is not absolute

So far we’ve been discussing relativity according to Galileo and

Newton, but there is also relativity according to Einstein. When

Einstein first began to develop the theory of relativity, around 1905,

the only real-world observations he could draw on were ambiguous

and indirect. Today, the evidence is part of everyday life. For ex-

ample, every time you use a GPS receiver, a, you’re using Einstein’s

a / This

Global

Positioning

theory of relativity. Somewhere between 1905 and today, technology

System (GPS) system, running

became good enough to allow conceptually simple experiments that

on a smartphone attached to a

students in the early 20th century could only discuss in terms like

bike’s handlebar,

depends on

“Imagine that we could. . . ” A good jumping-on point is 1971. In

Einstein’s

theory

of

relativity.

that year, J.C. Hafele and R.E. Keating, shown in the photo above,

Time flows at a different rates

aboard a GPS satellite than it

brought atomic clocks aboard commercial airliners, and went around

does on the bike, and the GPS

the world, once from east to west and once from west to east. The

software has to take this into

clocks were capable of keeping time to within a few nanoseconds.

account.

(A nanosecond, abbreviated ns, is one billionth of a second.) Hafele

71

and Keating observed that there was a discrepancy between the

times measured by the traveling clocks and the times measured by

similar clocks that stayed home at the U.S. Naval Observatory in

Washington. The east-going clock lost time, ending up off by −59

nanoseconds, while the west-going one gained 273 ns.

Causality

It reassuring that the effects on time were small compared to

the three-day lengths of the plane trips. There was therefore no

opportunity for paradoxical scenarios such as one in which the east-

going experimenter arrived back in Washington before he left and

then convinced himself not to take the trip. A theory that maintains

this kind of orderly relationship between cause and effect is said to

b / All three clocks are mov-

satisfy causality.

ing to the east. Even though the

west-going plane is moving to the

Time affected by motion and gravity

west relative to the air, the air

Hafele and Keating were testing specific quantitative predictions

is moving to the east due to the

earth’s rotation.

of relativity, and they verified them to within their experiment’s

error bars. Let’s work backward instead, and inspect the empirical

results for clues as to how time works.

The two traveling clocks experienced effects in opposite direc-

tions, and this suggests that the rate at which time flows depends

on the motion of the observer. The east-going clock was moving in

the same direction as the earth’s rotation, so its velocity relative to

the earth’s center was greater than that of the clock that remained

in Washington, while the west-going clock’s velocity was correspond-

ingly reduced. The fact that the east-going clock fell behind, and

the west-going one got ahead, shows that the effect of motion is to

make time go more slowly. This effect of motion on time was pre-

dicted by Einstein in his original 1905 paper on relativity, written

when he was 26.

If this had been the only effect in the Hafele-Keating experiment,

then we would have expected to see effects on the two flying clocks

that were equal in size. In fact, the two effects are unequal in size:

−59 ns and 273 ns. This turns out to be because there was a second

effect involved, a gravitational one, simply due to the planes’ being

up in the air. The gravitational effects are beyond the scope of this

book.

c / The

correspondence

prin-

ciple requires that the relativistic

The correspondence principle

distortion of time become small

The effects that Hafele and Keating observed were small. This

for small velocities. The effects

makes sense: the version of relativity worked out by Galileo (sec-

are so small that we have to

tions 2.2 and 2.3, pp. 39-53) had already been thoroughly tested by

describe them in scientific nota-

tion (p. 12). For example, 10−15

experiments under a wide variety of conditions, so a new theory like

means

0.000000000000001,

Einstein’s relativity must agree with Galileo’s to a good approxima-

which is a hundred thousand

tion, within the Galilean theory’s realm of applicability. This is an

times smaller than 10−10.

example of the correspondence principle (p. 31). The behavior of

72

Chapter 4

Relativity

the three clocks in the Hafele-Keating experiment shows that the

amount of time distortion increases as the speed of the clock’s mo-

tion increases. Newton lived in an era when the fastest mode of

transportation was a galloping horse, and the best pendulum clocks

would accumulate errors of perhaps a minute over the course of sev-

eral days. A horse is much slower than a jet plane, so the distortion

of time would have had a relative size of only ∼ 10−15 — much

smaller than the clocks were capable of detecting. At the speed of a

passenger jet, the effect is about 10−12, and state-of-the-art atomic

clocks in 1971 were capable of measuring that. A GPS satellite trav-

els much faster than a jet airplane, and the effect on the satellite

d / Two

events

are

given

as

turns out to be ∼ 10−10. The general idea here is that all physical

points on a graph of position

laws are approximations, and approximations aren’t simply right or

versus time. Joan of Arc helps to

wrong in different situations. Approximations are better or worse

restore Charles VII to the throne.

At a later time and a different

in different situations, and the question is whether a particular ap-

position, Joan of Arc is sentenced

proximation is good enough in a given situation to serve a particular

to death.

purpose. The faster the motion, the worse the Newtonian approxi-

mation of absolute time. Whether the approximation is good enough

depends on what you’re trying to accomplish. The correspondence

principle says that the approximation must have been good enough

to explain all the experiments done in the centuries before Einstein

came up with relativity.

By the way, don’t get an inflated idea of the importance of the

Hafele-Keating experiment. Relativity had already been confirmed

by a vast and varied body of experiments decades before 1971. The

only reason I’m giving such a prominent role to this experiment is

that it is conceptually very direct.

e / A change of units distorts

Distortion of time and space

an x -t graph. This graph depicts

Relativity says that when two observers are in different frames of

exactly the same events as figure

reference, each observer considers the other one’s perception of time

d. The only change is that the x

to be distorted. We’ll also see that something similar happens to

and t coordinates are measured

their observations of distances, so both space and time are distorted.

using different units, so the grid is

compressed in t and expanded in

What exactly is this distortion? How do we even conceptualize it?

x .

The idea isn’t really as radical as it might seem at first. We can

visualize the structure of space and time using a graph with position

and time on its axes. These graphs were introduced on p. 43 in fig-

ures d and e, but we’re going to look at them in a slightly different

way. Before, we used them to describe the motion of objects. The

grid underlying the graph was merely the stage on which the actors

played their parts. Now the background comes to the foreground:

it’s time and space themselves that we’re studying. We don’t nec-

essarily need to have a line or a curve drawn on top of the grid to

represent a particular object. We may, for example, just want to

f / A

convention

we’ll

use

to

talk about events, depicted as points on the graph as in figure d.

represent a distortion of time and

A distortion of the Cartesian grid underlying the graph can arise

space.

for perfectly ordinary reasons that Isaac Newton would have read-

Section 4.1

Relativity according to Einstein

73

ily accepted. For example, we can simply change the units used to

measure time and position, as in figure e.

We’re going to have quite a few examples of this type, so I’ll

adopt the convention shown in figure f for depicting them; this con-

vention was originally introduced in figure e on p. 43.

Figure f

summarizes the relationship between figures d and e in a more com-

pact form. The gray rectangle represents the original coordinate

g / A Galilean version of the

grid of figure d, while the grid of black lines represents the new ver-

relationship between two frames

sion from figure e. Omitting the grid from the gray rectangle makes

of reference, as introduced in

the diagram easier to decode visually.

figure e, p. 43.

As in all such

graphs in this chapter, the original

Our goal of unraveling the mysteries of special relativity amounts

coordinates, represented by the

to nothing more than finding out how to draw a diagram like f

gray rectangle, have a time axis

in the case where the two different sets of coordinates represent

that goes to the right, and a

measurements of time and space made by two different observers,

distance axis that goes straight

each in motion relative to the other. Galileo and Newton thought

up.

they knew the answer to this question, but their answer turned

out to be only approximately right. To avoid repeating the same

mistakes, we need to clearly spell out what we think are the basic

properties of time and space that will be a reliable foundation for

our reasoning.

Experiments show that:

1. The laws of physics have translation symmetry (section 2.1),

time symmetry (section 1.6), and rotational symmetry (p. 15

and section 3.3).

2. The principle of inertia holds (p. 16).

3. Causality holds, in the sense described on page 72.

4. Time depends on the state of motion of the observer.

If it were not for property 4, we could imagine that figure g

would give the correct transformation between frames of reference

in motion relative to one another. Let’s say that observer 1, whose

grid coincides with the gray rectangle, is a hitch-hiker standing by

the side of a road. Event A is a raindrop hitting his head, and

74

Chapter 4

Relativity

event B is another raindrop hitting his head. He says that A and B

occur at the same location in space. Observer 2 is a motorist who

drives by without stopping; to him, the passenger compartment of

his car is at rest, while the asphalt slides by underneath. He says

that A and B occur at different points in space, because during the

time between the first raindrop and the second, the hitch-hiker has

moved backward. On the other hand, observer 2 says that events A

and C occur in the same place, while the hitch-hiker disagrees. The

slope of the grid-lines is simply the velocity of the relative motion

of each observer relative to the other. (Recall that slope is defined

as the rise over the run. On these graphs of distance versus time,

the slope is the distance traveled divided by the elapsed time.)

h / In the units that are most

convenient for relativity, the trans-

Figure g has familiar, comforting, and eminently sensible behav-

formation has symmetry about a

ior, but it also happens to be wrong, because it violates property

45-degree diagonal line.

4. The distortion of the coordinate grid has only moved the vertical

lines up and down, so both observers agree that events like B and

C are simultaneous. If this was really the way things worked, then

all observers could synchronize all their clocks with one another for

once and for all, and the clocks would never get out of sync. This

contradicts the results of the Hafele-Keating experiment, in which

all three clocks were initially synchronized in Washington, but later

went out of sync because of their different states of motion.

Based on properties 1-4, there is only one possible way to mod-

ify g, which is the one shown in h.1 This distortion is the one that

Einstein predicted in 1905, and is known as the Lorentz transforma-

tion, after Hendrik Lorentz (1853-1928). The distortion is a kind of

smooshing and stretching, as suggested by the hands. Also, we’ve

already seen in figures d-f on page 73 that we’re free to stretch or

i / Interpretation of the Lorentz

compress everything as much as we like in the horizontal and verti-

transformation.

The slope in-

cal directions, because this simply corresponds to choosing different

dicated in the figure gives the

units of measurement for time and distance. In figure h I’ve chosen

relative velocity of the two frames

of reference. Events A and B that

units that give the whole drawing a convenient symmetry about a

were simultaneous in frame 1

45-degree diagonal line. Ordinarily it wouldn’t make sense to talk

are not simultaneous in frame 2,

about a 45-degree angle on a graph whose axes had different units.

where event A occurs to the right

But in relativity, the symmetric appearance of the transformation

of the t = 0 line represented by

tells us that space and time ought to be treated on the same footing,

the left edge of the grid, but event

and measured in the same units.

B occurs to its left.

The exact size and shape of the parallelogram are controlled by

the requirements that (i) the slope labeled in the figure corresponds

properly to the velocity; (ii) the units are the special ones described

1For a proof that no other version is possible, see ch. 23 of my free online

book Light and Matter.

Section 4.1

Relativity according to Einstein

75

above; and (iii) the area of the parallelogram is the same as the area of the original square.2

2The equal-area property is proved in Light and Matter.

76

Chapter 4

Relativity

The G factor

We’ve seen the experimental evidence that motion changes the

rate of flow of time, and this effect is correctly reproduced by the

Lorentz transformation.

Time dilation

A clock runs fastest in the frame of reference of an observer

who is at rest relative to the clock.

We define the factor G (Greek letter gamma) as in figure j. An

observer in motion relative to the clock at speed v perceives the

clock as running more slowly by a factor of G. For example, if G

equals 2, then the observer says the clock runs at half its normal

speed.

Figure k shows an example of how we can use properties (i)-(iii)

on p. 75 to find the value of G for a given velocity v of the clock

j / The G factor.

and the observer relative to one another. By plotting many such

points,3 we get the graph shown in figure l.

k / An

example

in

which

the

G factor is numerically simple to

work out. The bottom edge of the

parallelogram rises 3 units and

goes 5 units to the right, so its

slope is 3/5.

This slope repre-

sents the speed of one frame of

reference relative to the other. It’s

easy to verify that the square and

the parallelogram have the same

area, because one diagonal of

the square has been stretched to

twice its original length, the other

smooshed down by a half. The

l / The behavior of the γ factor.

(The velocity is in the special units

bottom-right corner of the square

described on p. 75. More on these units in section 4.2.)

is at a time of 4 units, while

the corresponding corner of the

parallelogram is at 5. As defined

For small velocities, the graph is nearly flat at γ ≈ 1, meaning that

in figure j, the ratio of these times

3

is the value of G = 5/4.

To avoid the tedious work of drawing many figures like k, one can use algebra

√

and geometry to derive the equation G = 1/ 1 − v2.

Section 4.1

Relativity according to Einstein

77

there is very little time dilation. This is required by the correspon-

dence principle.

Distances are also distorted:

Length contraction

A meter-stick appears longest to an observer who is at rest

relative to it. An observer moving relative to the meter-stick

at v observes the stick to be shortened by a factor of G.

m / Example 1: In the garage’s frame of reference, the bus is moving, and can fit in the garage due to its length contraction. In the bus’s frame of reference, the garage is moving, and can’t hold the bus due to its length contraction.

The garage paradox

example 1

One of the most famous of all the so-called relativity paradoxes

has to do with our incorrect feeling that simultaneity is well de-

fined. The idea is that one could take a schoolbus and drive it at

relativistic speeds into a garage of ordinary size, in which it nor-

mally would not fit. Because of the length contraction, the bus

would supposedly fit in the garage. The driver, however, will per-

ceive the garage as being contracted and thus even less able to

contain the bus.

The paradox is resolved when we recognize that the concept of

fitting the bus in the garage “all at once” contains a hidden as-

78

Chapter 4

Relativity

sumption, the assumption that it makes sense to ask whether the

front and back of the bus can simultaneously be in the garage.

Observers in different frames of reference moving at high relative

speeds do not necessarily agree on whether things happen si-

multaneously. As shown in figure m, the person in the garage’s

frame can shut the door at an instant B he perceives to be si-

multaneous with the front bumper’s arrival A at the back wall of

the garage, but the driver would not agree about the simultaneity

of these two events, and would perceive the door as having shut

long after she plowed through the back wall.

Section 4.1

Relativity according to Einstein

79

4.2 Speeds in relativity

The universal speed c

Let’s think a little more about the role of the 45-degree diagonal

in the Lorentz transformation. Slopes on these graphs are inter-

preted as velocities. This line has a slope of 1 in our special rel-

ativistic units, but that slope corresponds to some number, call it

c, in ordinary units of meters per second. Now note what happens

when we perform a Lorentz transformation: this particular line gets

stretched, but the new version of the line lies right on top of the old

one, and its slope stays the same. In other words, if one observer

says that something has a velocity equal to c, every other observer

will agree on that velocity as well.

Velocities don’t simply add and subtract.

This is counterintuitive, since we expect velocities to add and

n / A proof that causality im-

subtract in relative motion. If a dog is running away from me at 5

poses a universal speed limit. In

the original frame of reference,

m/s relative to the sidewalk, and I run after it at 3 m/s, the dog’s

represented by the square, event

velocity in my frame of reference is 2 m/s. According to everything

A happens a little before event B.

we have learned about motion, the dog must have different speeds

In the new frame, shown by the

in the two frames: 5 m/s in the sidewalk’s frame and 2 m/s in

parallelogram, A happens after

mine. But velocities are measured by dividing a distance by a time,

t = 0, but B happens before t = 0;

and both distance and time are distorted by relativistic effects, so

that is, B happens before A. The

we actually shouldn’t expect the ordinary arithmetic addition of

time ordering of the two events

has been reversed. This can only

velocities to hold in relativity; it’s an approximation that’s valid at

happen because events A and B

velocities that are small compared to c.

are very close together in time

and fairly far apart in space. The

A universal speed limit

line segment connecting A and

For example, suppose Janet takes a trip in a spaceship, and

B has a slope greater than 1,

accelerates until she is moving at 0.6c relative to the earth. She

meaning that if we wanted to be

then launches a space probe in the forward direction at a speed

present at both events, we would

relative to her ship of 0.6c. We might think that the probe was then

have to travel at a speed greater

than c (which equals 1 in the

moving at a velocity of 1.2c, but in fact the answer is still less than

units used on this graph). You will

c (problem 1, page 89). This is an example of a more general fact

find that if you pick any two points

about relativity, which is that c represents a universal speed limit.

for which the slope of the line

This is required by causality, as shown in figure n.

segment connecting them is less

than 1, you can never get them to

Light travels at c.

straddle the new t = 0 line in this

Now consider a beam of light. We’re used to talking casually

funny, time-reversed way. Since

different observers disagree on

about the “speed of light,” but what does that really mean? Motion

the time order of events like A

is relative, so normally if we want to talk about a velocity, we have

and B, causality requires that

to specify what it’s measured relative to. A sound wave has a certain

information

never

travel

from

speed relative to the air, and a water wave has its own speed relative

A to B or from B to A; if it did,

to the water. If we want to measure the speed of an ocean wave, for

then we would have time-travel

example, we should make sure to measure it in a frame of reference

paradoxes. The conclusion is that

at rest relative to the water. But light isn’t a vibration of a physical

c is the maximum speed of cause

and effect in relativity.

medium; it can propagate through the near-perfect vacuum of outer

space, as when rays of sunlight travel to earth. This seems like a

80

Chapter 4

Relativity

paradox: light is supposed to have a specific speed, but there is no

way to decide what frame of reference to measure it in. The way

out of the paradox is that light must travel at a velocity equal to c.

Since all observers agree on a velocity of c, regardless of their frame

of reference, everything is consistent.

The Michelson-Morley experiment

The constancy of the speed of light had in fact already been

observed when Einstein was an 8-year-old boy, but because nobody

could figure out how to interpret it, the result was largely ignored.

In 1887 Michelson and Morley set up a clever apparatus to measure

any difference in the speed of light beams traveling east-west and

north-south. The motion of the earth around the sun at 110,000

km/hour (about 0.01% of the speed of light) is to our west during the

day. Michelson and Morley believed that light was a vibration of a

mysterious medium called the ether, so they expected that the speed

of light would be a fixed value relative to the ether. As the earth

moved through the ether, they thought they would observe an effect

on the velocity of light along an east-west line. For instance, if they

released a beam of light in a westward direction during the day, they

expected that it would move away from them at less than the normal

speed because the earth was chasing it through the ether. They were

surprised when they found that the expected 0.01% change in the

speed of light did not occur.

The ring laser gyroscope

example 2

If you’ve flown in a jet plane, you can thank relativity for help-

ing you to avoid crashing into a mountain or an ocean. Figure o

shows a standard piece of navigational equipment called a ring

laser gyroscope. A beam of light is split into two parts, sent

around the perimeter of the device, and reunited. Since the speed

of light is constant, we expect the two parts to come back together

at the same time. If they don’t, it’s evidence that the device has

been rotating. The plane’s computer senses this and notes how

much rotation has accumulated.

No frequency-dependence

example 3

Relativity has only one universal speed, so it requires that all light

waves travel at the same speed, regardless of their frequency

and wavelength. Presently the best experimental tests of the in-

variance of the speed of light with respect to wavelength come

from astronomical observations of gamma-ray bursts, which are

sudden outpourings of high-frequency light, believed to originate

from a supernova explosion in another galaxy. One such obser-

vation, in 2009,4 found that the times of arrival of all the different

frequencies in the burst differed by no more than 2 seconds out

of a total time in flight on the order of ten billion years!

o / A

ring

laser

gyroscope.

4http://arxiv.org/abs/0908.1832

Section 4.2

Speeds in relativity

81

An interstellar road trip

example 4

Because the distances between the stars are so vast, it’s con-

venient to measure them in light-years rather than kilometers. A

light-year is defined as the distance traveled by light in one year.

If we adopt the year as our unit of time, and the light-year as our

unit of distance, then the speed of light is 1, i.e., these units qual-

ify as the kind of “special units” that we’ve been assuming in all

the graphs.

Suppose that Alice stays on earth while her twin Betty heads off

in a spaceship for Tau Ceti, a nearby star. Tau Ceti is 12 light-

years away, so even though Betty travels at 87% of the speed of

light, it will take her a long time to get there: 14 years, according

to Alice.

p / Example 4.

Betty experiences time dilation. At this speed, her γ is 2.0, so that

the voyage will only seem to her to last 7 years. But there is per-

fect symmetry between Alice’s and Betty’s frames of reference, so

Betty agrees with Alice on their relative speed; Betty sees herself

as being at rest, while the sun and Tau Ceti both move backward

at 87% of the speed of light. How, then, can she observe Tau Ceti

to get to her in only 7 years, when it should take 14 years to travel

12 light-years at this speed?

We need to take into account length contraction. Betty sees the

distance between the sun and Tau Ceti to be shrunk by a factor of

2. The same thing occurs for Alice, who observes Betty and her

spaceship to be foreshortened.

82

Chapter 4

Relativity

Discussion questions

A

A person in a spaceship moving at 99.99999999% of the speed

of light relative to Earth shines a flashlight forward through dusty air, so

the beam is visible. What does she see? What would it look like to an

observer on Earth?

B

A question that students often struggle with is whether time and

space can really be distorted, or whether it just seems that way. Compare

with optical illusions or magic tricks. How could you verify, for instance,

that the lines in the figure are actually parallel? Are relativistic effects the

same, or not?

C

On a spaceship moving at relativistic speeds, would a lecture seem

even longer and more boring than normal?

Discussion question B

D

Mechanical clocks can be affected by motion. For example, it was

a significant technological achievement to build a clock that could sail

aboard a ship and still keep accurate time, allowing longitude to be deter-

mined. How is this similar to or different from relativistic time dilation?

E

The figure shows an artist’s rendering of the length contraction for the

collision of two gold nuclei at relativistic speeds in the RHIC accelerator in

Long Island, New York, which went on line in 2000. The gold nuclei would

appear nearly spherical (or just slightly lengthened like an American foot-

ball) in frames moving along with them, but in the laboratory’s frame, they

both appear drastically foreshortened as they approach the point of col-

lision. The later pictures show the nuclei merging to form a hot soup, in

which experimenters hope to observe a new form of matter. What would

the shapes of the two nuclei look like to a microscopic observer riding on

the left-hand nucleus? To an observer riding on the right-hand one? Can

they agree on what is happening? If not, why not — after all, shouldn’t

they see the same thing if they both compare the two nuclei side-by-side

at the same instant in time?

Discussion question E: colliding nuclei show relativistic length contrac-

tion.

Section 4.2

Speeds in relativity

83

F

If you stick a piece of foam rubber out the window of your car while

driving down the freeway, the wind may compress it a little. Does it make

sense to interpret the relativistic length contraction as a type of strain

that pushes an object’s atoms together like this? How does this relate to

discussion question E?

G

The machine-gunner in the figure sends out a spray of bullets.

Suppose that the bullets are being shot into outer space, and that the

distances traveled are trillions of miles (so that the human figure in the

diagram is not to scale). After a long time, the bullets reach the points

shown with dots which are all equally far from the gun. Their arrivals

at those points are events A through E, which happen at different times.

Sketch these events on a position-time graph. The chain of impacts ex-

tends across space at a speed greater than c. Does this violate special

relativity?

Discussion question G.

84

Chapter 4

Relativity

4.3 Dynamics

So far we have said nothing about how to predict motion in relativ-

ity. Do Newton’s laws still work? Do conservation laws still apply?

The answer is yes, but many of the definitions need to be modified,

and certain entirely new phenomena occur, such as the conversion

of mass to energy and energy to mass, as described by the famous

equation E = mc2.

Momentum

Here’s a flawed scheme for traveling faster than the speed of

light. The basic idea can be demonstrated by dropping a ping-pong

ball and a baseball stacked on top of each other like a snowman.

They separate slightly in mid-air, and the baseball therefore has

time to hit the floor and rebound before it collides with the ping-

pong ball, which is still on the way down. The result is a surprise

if you haven’t seen it before: the ping-pong ball flies off at high

speed and hits the ceiling! A similar fact is known to people who

investigate the scenes of accidents involving pedestrians. If a car

moving at 90 kilometers per hour hits a pedestrian, the pedestrian

flies off at nearly double that speed, 180 kilometers per hour. Now

suppose the car was moving at 90 percent of the speed of light.

Would the pedestrian fly off at 180% of c?

r / An unequal collision, viewed in

the center-of-mass frame, 1, and

in the frame where the small ball

is initially at rest, 2. The motion

is shown as it would appear on

the film of an old-fashioned movie

camera, with an equal amount of

time separating each frame from

the next.

Film 1 was made by

a camera that tracked the center

of mass, film 2 by one that was

initially tracking the small ball,

and kept on moving at the same

speed after the collision.

To see why not, we have to back up a little and think about

where this speed-doubling result comes from. For any collision, there

is a special frame of reference, the center-of-mass frame, in which

the two colliding objects approach each other, collide, and rebound

with their velocities reversed. In the center-of-mass frame, the total

Section 4.3

Dynamics

85

momentum of the objects is zero both before and after the collision.

Figure r/1 shows such a frame of reference for objects of very

unequal mass. Before the collision, the large ball is moving relatively

slowly toward the top of the page, but because of its greater mass,

its momentum cancels the momentum of the smaller ball, which is

moving rapidly in the opposite direction. The total momentum is

zero. After the collision, the two balls just reverse their directions of

motion. We know that this is the right result for the outcome of the

collision because it conserves both momentum and kinetic energy,

and everything not forbidden is compulsory, i.e., in any experiment,

there is only one possible outcome, which is the one that obeys all

the conservation laws.

self-check A

How do we know that momentum and kinetic energy are conserved in

figure r/1?

. Answer, p. 179

Let’s make up some numbers as an example. Say the small ball

has a mass of 1 kg, the big one 8 kg. In frame 1, let’s make the

velocities as follows:

before the collision

after the collision

-0.8

0.8

0.1

-0.1

Figure r/2 shows the same collision in a frame of reference where

the small ball was initially at rest. To find all the velocities in this

frame, we just add 0.8 to all the ones in the previous table.

before the collision

after the collision

0

1.6

0.9

0.7

In this frame, as expected, the small ball flies off with a velocity,

1.6, that is almost twice the initial velocity of the big ball, 0.9.

If all those velocities were in meters per second, then that’s ex-

actly what happened. But what if all these velocities were in units

of the speed of light? Now it’s no longer a good approximation just

to add velocities. We need to combine them according to the rela-

tivistic rules. For instance, reasoning similar to that in problem 1,

page 89 tells us that combining a velocity of 0.8 times the speed of

light with another velocity of 0.8 results in 0.98, not 1.6. The results

are very different:

before the collision

after the collision

0

0.98

0.83

0.76

We can interpret this as follows. Figure r/1 is one in which the

big ball is moving fairly slowly. This is very nearly the way the

scene would be seen by an ant standing on the big ball. According

86

Chapter 4

Relativity

s / An 8-kg ball moving at 83%

of the speed of light hits a 1-kg

ball. The balls appear foreshort-

ened due to the relativistic distor-

tion of space.

to an observer in frame s, however, both balls are moving at nearly

the speed of light after the collision.

Because of this, the balls

appear foreshortened, but the distance between the two balls is also

shortened. To this observer, it seems that the small ball isn’t pulling

away from the big ball very fast.

Now here’s what’s interesting about all this. The outcome shown

in figure r/2 was supposed to be the only one possible, the only

one that satisfied both conservation of energy and conservation of

momentum. So how can the different result shown in figure s be

possible? The answer is that relativistically, momentum must not

equal mv. The old, familiar definition is only an approximation

that’s valid at low speeds. If we observe the behavior of the small

ball in figure s, it looks as though it somehow had some extra inertia.

It’s as though a football player tried to knock another player down

without realizing that the other guy had a three-hundred-pound bag

full of lead shot hidden under his uniform — he just doesn’t seem

to react to the collision as much as he should. This extra inertia is

described by redefining momentum as

momentum = mγv.

At very low velocities, γ is close to 1, and the result is very nearly

mv, as demanded by the correspondence principle.

But at very

high velocities, γ gets very big — the small ball in figure s has a

γ of 5.0, and therefore has five times more inertia than we would

expect nonrelativistically.

Equivalence of mass and energy

Now we’re ready to see why mass and energy must be equivalent

as claimed in the famous E = mc2. So far we’ve only considered

collisions in which none of the kinetic energy is converted into any

other form of energy, such as heat or sound. Let’s consider what

happens if a blob of putty moving at velocity v hits another blob

that is initially at rest, sticking to it. The nonrelativistic result is

that to obey conservation of momentum the two blobs must fly off

together at v/2. Half of the initial kinetic energy has been converted

to heat.5

5A double-mass object moving at half the speed does not have the same

kinetic energy. Kinetic energy depends on the square of the velocity, so cutting

Section 4.3

Dynamics

87

Relativistically, however, an interesting thing happens. A hot

object has more momentum than a cold object! This is because

the relativistically correct expression for momentum is mγv, and

the more rapidly moving atoms in the hot object have higher values

of γ. In our collision, the final combined blob must therefore be

moving a little more slowly than the expected v/2, since otherwise

the final momentum would have been a little greater than the initial

momentum. To an observer who believes in conservation of momen-

tum and knows only about the overall motion of the objects and not

about their heat content, the low velocity after the collision would

seem to be the result of a magical change in the mass, as if the mass

of two combined, hot blobs of putty was more than the sum of their

individual masses.

Now we know that the masses of all the atoms in the blobs must

be the same as they always were. The change is due to the change in

γ with heating, not to a change in mass. The heat energy, however,

seems to be acting as if it was equivalent to some extra mass.

But this whole argument was based on the fact that heat is a

form of kinetic energy at the atomic level. Would E = mc2 apply to

other forms of energy as well? Suppose a rocket ship contains some

electrical energy stored in a battery. If we believed that E = mc2

applied to forms of kinetic energy but not to electrical energy, then

we would have to believe that the pilot of the rocket could slow

the ship down by using the battery to run a heater! This would

not only be strange, but it would violate the principle of relativity,

because the result of the experiment would be different depending

on whether the ship was at rest or not. The only logical conclusion is

that all forms of energy are equivalent to mass. Running the heater

then has no effect on the motion of the ship, because the total

energy in the ship was unchanged; one form of energy (electrical)

was simply converted to another (heat).

The equation E = mc2 tells us how much energy is equivalent

to how much mass: the conversion factor is the square of the speed

of light, c. Since c a big number, you get a really really big number

when you multiply it by itself to get c2. This means that even a small

amount of mass is equivalent to a very large amount of energy.

We’ve already seen several examples of applications of E = mc2,

on page 30.

the velocity in half reduces the energy by a factor of 1/4, which, multiplied by

the doubled mass, makes 1/2 the original energy.

88

Chapter 4

Relativity

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

The figure illustrates a Lorentz transformation using the con-

ventions described on p. 43. For simplicity, the transformation cho-

sen is one that lengthens one diagonal by a factor of 2. Since Lorentz

transformations preserve area, the other diagonal is shortened by a

factor of 2. Let the original frame of reference, depicted with the

square, be A, and the new one B. (a) By measuring with a ruler on

the figure, show that the velocity of frame B relative to frame A is

0.6c. (b) Print out a copy of the page. With a ruler, draw a third

parallelogram that represents a second successive Lorentz transfor-

mation, one that lengthens the long diagonal by another factor of

2. Call this third frame C. Use measurements with a ruler to deter-

mine frame C’s velocity relative to frame A. Does it equal double

the velocity found in part a? Explain why it should be expected to

√

turn out the way it does.

Problems

89

2

Astronauts in three different spaceships are communicating

with each other. Those aboard ships A and B agree on the rate at

which time is passing, but they disagree with the ones on ship C.

(a) Alice is aboard ship A. How does she describe the motion of her

own ship, in its frame of reference?

(b) Describe the motion of the other two ships according to Alice.

(c) Give the description according to Betty, whose frame of reference

is ship B.

(d) Do the same for Cathy, aboard ship C.

3

Figure e on p. 43 shows a convention for representing a Lorentz

transformation using a parallelogram. Recall that on these graphs,

the slope of the parallelogram’s bottom edge represents the velocity,

and that special units are assumed in which the speed of light equals

1. What would happen to the diagram if the velocity equaled the

speed of light?

Problem 4.

4

The figure shows a famous thought experiment devised by

Einstein. A train is moving at constant velocity to the right when

bolts of lightning strike the ground near its front and back. Alice,

standing on the dirt at the midpoint of the flashes, observes that

the light from the two flashes arrives simultaneously, so she says the

two strikes must have occurred simultaneously. Bob, meanwhile,

is sitting aboard the train, at its middle. He passes by Alice at

the moment when Alice later figures out that the flashes happened.

Later, he receives flash 2, and then flash 1. He infers that since

both flashes traveled half the length of the train, flash 2 must have

occurred first. How can this be reconciled with Alice’s belief that

the flashes were simultaneous? Explain using a graph. Note that

the light from the flashes will move at velocity c or −c, represented

by lines at 45-degree angles.

5

The rod in the figure is perfectly rigid. At event A, the

hammer strikes one end of the rod. At event B, the other end moves.

Since the rod is perfectly rigid, it can’t compress, so A and B are

simultaneous. In frame 2, B happens before A. Did the motion at

Problem 5.

the right end cause the person on the left to decide to pick up the

hammer and use it?

90

Chapter 4

Relativity

6

Suppose that the starship Enterprise from Star Trek has a

mass of 8.0 × 107 kg, about the same as the Queen Elizabeth 2.

Suppose that it was moving at half the speed of light. Read its

G off of the graph in figure l on p. 77, and use this to compute

its energy. Compare with the total energy content of the world’s

√

nuclear arsenals, which is about 1021 J.

7

In the graph in figure l on p. 77, the G factor blows up to infinity

as the velocity approaches the speed of light. Recall that force is

the rate of change of momentum, and that relativistic momentum

is given by mγv. Based on these ideas, what would happen if we

applied a constant force to an object for a very long time? Would

it eventually go faster than the speed of light?

Problems

91

92

Chapter 4

Relativity

Chapter 5

Electricity

Where the telescope ends, the microscope begins. Which of the two

has the grander view?

Victor Hugo

His father died during his mother’s pregnancy. Rejected by her

as a boy, he was packed off to boarding school when she remarried.

He himself never married, but in middle age he formed an intense

relationship with a much younger man, a relationship that he ter-

minated when he underwent a psychotic break. Following his early

scientific successes, he spent the rest of his professional life mostly

in frustration over his inability to unlock the secrets of alchemy.

The man being described is Isaac Newton, but not the triumphant

Newton of the standard textbook hagiography. Why dwell on the

sad side of his life? To the modern science educator, Newton’s life-

long obsession with alchemy may seem an embarrassment, a distrac-

tion from his main achievement, the creation of the modern science

of mechanics. To Newton, however, his alchemical researches were

naturally related to his investigations of force and motion. What

was radical about Newton’s analysis of motion was its universal-

ity: it succeeded in describing both the heavens and the earth with

the same equations, whereas previously it had been assumed that

the sun, moon, stars, and planets were fundamentally different from

earthly objects. But Newton realized that if science was to describe

all of nature in a unified way, it was not enough to unite the human

scale with the scale of the universe: he would not be satisfied until

93

he fit the microscopic universe into the picture as well.

It should not surprise us that Newton failed. Although he was a

firm believer in the existence of atoms, there was no more experimen-

tal evidence for their existence than there had been when the ancient

Greeks first posited them on purely philosophical grounds. Alchemy

labored under a tradition of secrecy and mysticism. Newton had

already almost single-handedly transformed the fuzzyheaded field

of “natural philosophy” into something we would recognize as the

modern science of physics, and it would be unjust to criticize him

for failing to change alchemy into modern chemistry as well. The

time was not ripe. The microscope was a new invention, and it was

cutting-edge science when Newton’s contemporary Hooke discovered

that living things were made out of cells.

5.1 The quest for the atomic force

Newton was not the first of the age of reason. He was the last of

the magicians.

John Maynard Keynes

Nevertheless it will be instructive to pick up Newton’s train of

thought and see where it leads us with the benefit of modern hind-

sight. In uniting the human and cosmic scales of existence, he had

reimagined both as stages on which the actors were objects (trees

and houses, planets and stars) that interacted through attractions

and repulsions. He was already convinced that the objects inhab-

iting the microworld were atoms, so it remained only to determine

what kinds of forces they exerted on each other.

His next insight was no less brilliant for his inability to bring it to

fruition. He realized that the many human-scale forces — friction,

sticky forces, the normal forces that keep objects from occupying

the same space, and so on — must all simply be expressions of a

more fundamental force acting between atoms. Tape sticks to paper

because the atoms in the tape attract the atoms in the paper. My

house doesn’t fall to the center of the earth because its atoms repel

the atoms of the dirt under it.

Here he got stuck. It was tempting to think that the atomic force

was a form of gravity, which he knew to be universal, fundamental,

and mathematically simple. Gravity, however, is always attractive,

so how could he use it to explain the existence of both attractive

and repulsive atomic forces? The gravitational force between ob-

jects of ordinary size is also extremely small, which is why we never

notice cars and houses attracting us gravitationally. It would be

hard to understand how gravity could be responsible for anything

as vigorous as the beating of a heart or the explosion of gunpowder.

Newton went on to write a million words of alchemical notes filled

with speculation about some other force, perhaps a “divine force” or

“vegetative force” that would for example be carried by the sperm

94

Chapter 5

Electricity

to the egg.

Luckily, we now know enough to investigate a different suspect

as a candidate for the atomic force: electricity. Electric forces are

often observed between objects that have been prepared by rubbing

(or other surface interactions), for instance when clothes rub against

each other in the dryer. A useful example is shown in figure 5.1/1:

stick two pieces of tape on a tabletop, and then put two more pieces

on top of them. Lift each pair from the table, and then separate

them.

The two top pieces will then repel each other, 5.1/2, as

will the two bottom pieces. A bottom piece will attract a top piece,

however, 5.1/3. Electrical forces like these are similar in certain ways

to gravity, the other force that we already know to be fundamental:

• Electrical forces are universal.

Although some substances,

such as fur, rubber, and plastic, respond more strongly to

electrical preparation than others, all matter participates in

electrical forces to some degree. There is no such thing as a

“nonelectric” substance.

Matter is both inherently gravita-

tional and inherently electrical.

a / Four

pieces

of

tape

are

prepared, 1, as described in the

• Experiments show that the electrical force, like the gravita-

text. Depending on which com-

tional force, is an inverse square force. That is, the electrical

bination is tested, the interaction

force between two spheres is proportional to 1/r2, where r is

can be either repulsive, 2, or

attractive, 3.

the center-to-center distance between them.

Furthermore, electrical forces make more sense than gravity as

candidates for the fundamental force between atoms, because we

have observed that they can be either attractive or repulsive.

5.2 Charge, electricity and magnetism

Charge

“Charge” is the technical term used to indicate that an object

participates in electrical forces. This is to be distinguished from

the common usage, in which the term is used indiscriminately for

anything electrical. For example, although we speak colloquially of

“charging” a battery, you may easily verify that a battery has no

charge in the technical sense, e.g., it does not exert any electrical

force on a piece of tape that has been prepared as described in

section 5.1.

Two types of charge

We can easily collect reams of data on electrical forces between

different substances that have been charged in different ways. We

find for example that cat fur prepared by rubbing against rabbit

fur will attract glass that has been rubbed on silk. How can we

make any sense of all this information?

A vast simplification is

Section 5.2

Charge, electricity and magnetism

95

achieved by noting that there are really only two types of charge.

Suppose we pick cat fur rubbed on rabbit fur as a representative of

type A, and glass rubbed on silk for type B. We will now find that

there is no “type C.” Any object electrified by any method is either

A-like, attracting things A attracts and repelling those it repels, or

B-like, displaying the same attractions and repulsions as B. The two

types, A and B, always display opposite interactions. If A displays

an attraction with some charged object, then B is guaranteed to

undergo repulsion with it, and vice-versa.

The coulomb

Although there are only two types of charge, each type can come

in different amounts.

The metric unit of charge is the coulomb

(rhymes with “drool on”), defined as follows:

One Coulomb (C) is the amount of charge such that a force of

9.0 × 109 N occurs between two pointlike objects with charges

of 1 C separated by a distance of 1 m.

The notation for an amount of charge is q. The numerical factor

in the definition is historical in origin, and is not worth memoriz-

ing. The definition is stated for pointlike, i.e., very small, objects,

because otherwise different parts of them would be at different dis-

tances from each other.

A model of two types of charged particles

Experiments show that all the methods of rubbing or otherwise

charging objects involve two objects, and both of them end up get-

ting charged. If one object acquires a certain amount of one type of

charge, then the other ends up with an equal amount of the other

type. Various interpretations of this are possible, but the simplest

is that the basic building blocks of matter come in two flavors, one

with each type of charge. Rubbing objects together results in the

transfer of some of these particles from one object to the other. In

this model, an object that has not been electrically prepared may ac-

tually possesses a great deal of both types of charge, but the amounts

are equal and they are distributed in the same way throughout it.

Since type A repels anything that type B attracts, and vice versa,

the object will make a total force of zero on any other object. The

rest of this chapter fleshes out this model and discusses how these

mysterious particles can be understood as being internal parts of

atoms.

Use of positive and negative signs for charge

Because the two types of charge tend to cancel out each other’s

forces, it makes sense to label them using positive and negative signs,

and to discuss the total charge of an object. It is entirely arbitrary

which type of charge to call negative and which to call positive.

Benjamin Franklin decided to describe the one we’ve been calling

96

Chapter 5

Electricity

“A” as negative, but it really doesn’t matter as long as everyone is

consistent with everyone else. An object with a total charge of zero

(equal amounts of both types) is referred to as electrically neutral.

self-check A

Criticize the following statement: “There are two types of charge, attrac-

tive and repulsive.”

. Answer, p.

180

Coulomb’s law

A large body of experimental observations can be summarized

as follows:

Coulomb’s law: The magnitude of the force acting between

pointlike charged objects at a center-to-center distance r is given

by the equation

|q

|

1||q2|

F| = k

,

r2

where the constant k equals 9.0 × 109 N·m2/C2. The force is attrac-

tive if the charges are of different signs, and repulsive if they have

the same sign.

Conservation of charge

An even more fundamental reason for using positive and negative

signs for electrical charge is that experiments show that with the

b / A

charged

piece

of

tape

signs defined this way, the total amount of charge is a conserved

attracts

uncharged

pieces

of

quantity. This is why we observe that rubbing initially uncharged

paper from a distance, and they

substances together always has the result that one gains a certain

leap up to it.

amount of one type of charge, while the other acquires an equal

amount of the other type. Conservation of charge seems natural in

our model in which matter is made of positive and negative particles.

If the charge on each particle is a fixed property of that type of

particle, and if the particles themselves can be neither created nor

destroyed, then conservation of charge is inevitable.

Electrical forces involving neutral objects

As shown in figure 5.2.3, an electrically charged object can at-

tract objects that are uncharged. How is this possible? The key is

that even though each piece of paper has a total charge of zero, it

has at least some charged particles in it that have some freedom to

move. Suppose that the tape is positively charged, 5.2.3. Mobile

particles in the paper will respond to the tape’s forces, causing one

end of the paper to become negatively charged and the other to

become positive. The attraction between the paper and the tape

c / The

paper

has

zero

total

is now stronger than the repulsion, because the negatively charged

charge, but it does have charged

end is closer to the tape.

particles in it that can move.

self-check B

Section 5.2

Charge, electricity and magnetism

97

What would have happened if the tape was negatively charged?

.

Answer, p. 180

The atom, and subatomic particles

I once had a student whose father had been an electrician. He

told me that his father had never really believed that an electrical

current in a wire could be carried by moving electrons, because the

wire was solid, and it seemed to him that physical particles moving

through it would eventually have drilled so many holes through it

d / Examples of the construction

that it would have crumbled. It may sound as though I’m trying

of atoms:

hydrogen (top) and

to make fun of the father, but actually he was behaving very much

helium (bottom).

On this scale,

like the model of the skeptical scientist: he didn’t want to make

the electrons’ orbits would be the

hypotheses that seemed more complicated than would be necessary

size of a college campus.

in order to explain his observations. Physicists before about 1905

were in exactly the same situation. They knew all about electrical

circuits, and had even invented radio, but knew absolutely nothing

about subatomic particles. In other words, it hardly ever matters

that electricity really is made of charged particles, and it hardly ever

matters what those particles are. Nevertheless, it may avoid some

confusion to give a brief review of how an atom is put together:

charge

mass in units of

location in atom

the proton’s mass

proton

+e

1

in nucleus

neutron

0

1.001

in nucleus

electron

−e

1/1836

orbiting nucleus

The symbol e in this table is an abbreviation for 1.60 × 10−19 C.

The physicist Robert Millikan discovered in 1911 that any material

object (he used oil droplets) would have a charge that was a multiple

of this number, and today we interpret that as being a consequence

of the fact that matter is made of atoms, and atoms are made of

particles whose charges are plus and minus this amount.

Electric current

If the fundamental phenomenon is the motion of charged parti-

cles, then how can we define a useful numerical measurement of it?

We might describe the flow of a river simply by the velocity of the

water, but velocity will not be appropriate for electrical purposes

because we need to take into account how much charge the moving

particles have, and in any case there are no practical devices sold

at Radio Shack that can tell us the velocity of charged particles.

Experiments show that the intensity of various electrical effects is

related to a different quantity: the number of coulombs of charge

that pass by a certain point per second. By analogy with the flow

of water, this quantity is called the electric current, I. Its units

e

of coulombs/second are more conveniently abbreviated as amperes,

/ Andr é Marie Amp ère (1775-

1836).

1 A=1 C/s. (In informal speech, one usually says “amps.”)

98

Chapter 5

Electricity

The main subtlety involved in this definition is how to account

for the two types of charge. The stream of water coming from a

hose is made of atoms containing charged particles, but it produces

none of the effects we associate with electric currents. For example,

you do not get an electrical shock when you are sprayed by a hose.

This type of experiment shows that the effect created by the motion

of one type of charged particle can be canceled out by the motion of

the opposite type of charge in the same direction. In water, every

oxygen atom with a charge of +8e is surrounded by eight electrons

with charges of −e, and likewise for the hydrogen atoms.

We therefore refine our definition of current as follows:

definition of electric current

When charged particles are exchanged between regions of space

A and B, the electric current flowing from A to B is

change in B’s charge

I =

,

t

where the transfer occurs over a period of time t.

In the garden hose example, your body picks up equal amounts of

positive and negative charge, resulting in no change in your total

charge, so the electrical current flowing into you is zero.

Ions moving across a cell membrane

example 1

. Figure f shows ions, labeled with their charges, moving in or

out through the membranes of four cells. If the ions all cross

the membranes during the same interval of time, how would the

currents into the cells compare with each other?

. Cell A has positive current going into it because its charge is

increased, i.e., has a positive change in its charge.

Cell B has the same current as cell A, because by losing one unit

of negative charge it also ends up increasing its own total charge

by one unit.

Cell C’s total charge is reduced by three units, so it has a large

negative current going into it.

Cell D loses one unit of charge, so it has a small negative current

into it.

f / Example 1

Section 5.2

Charge, electricity and magnetism

99

It may seem strange to say that a negatively charged particle

going one way creates a current going the other way, but this is

quite ordinary. As we will see, currents flow through metal wires

via the motion of electrons, which are negatively charged, so the

direction of motion of the electrons in a circuit is always opposite to

the direction of the current. Of course it would have been convenient

if Benjamin Franklin had defined the positive and negative signs of

charge the opposite way, since so many electrical devices are based

on metal wires.

Number of electrons flowing through a lightbulb

example 2

. If a lightbulb has 1.0 A flowing through it, how many electrons

will pass through the filament in 1.0 s?

. We are only calculating the number of electrons that flow, so we

can ignore the positive and negative signs. Solving for (charge) =

It gives a charge of 1.0 C flowing in this time interval. The number

of electrons is

electrons

number of electrons = coulombs × coulomb

coulombs

= coulombs/ electron

= 1.0 C/e

= 6.2 × 1018

That’s a lot of electrons!

5.3 Circuits

How can we put electric currents to work? The only method of

controlling electric charge we have studied so far is to charge differ-

ent substances, e.g., rubber and fur, by rubbing them against each

other. Figure g/1 shows an attempt to use this technique to light

a lightbulb. This method is unsatisfactory. True, current will flow

through the bulb, since electrons can move through metal wires, and

the excess electrons on the rubber rod will therefore come through

the wires and bulb due to the attraction of the positively charged

fur and the repulsion of the other electrons. The problem is that

after a zillionth of a second of current, the rod and fur will both have

run out of charge. No more current will flow, and the lightbulb will

go out.

Figure g/2 shows a setup that works. The battery pushes charge

through the circuit, and recycles it over and over again. (We will

have more to say later in this chapter about how batteries work.)

g / 1.

Static electricity runs

This is called a complete circuit. Today, the electrical use of the

out quickly. 2. A practical circuit.

word “circuit” is the only one that springs to mind for most people,

3. An open circuit. 4. How an

ammeter works.

5.

Measuring

but the original meaning was to travel around and make a round

the current with an ammeter.

trip, as when a circuit court judge would ride around the boondocks,

dispensing justice in each town on a certain date.

100

Chapter 5

Electricity

Note that an example like g/3 does not work. The wire will

quickly begin acquiring a net charge, because it has no way to get

rid of the charge flowing into it. The repulsion of this charge will

make it more and more difficult to send any more charge in, and

soon the electrical forces exerted by the battery will be canceled

out completely. The whole process would be over so quickly that

the filament would not even have enough time to get hot and glow.

This is known as an open circuit. Exactly the same thing would

happen if the complete circuit of figure g/2 was cut somewhere with

a pair of scissors, and in fact that is essentially how an ordinary

light switch works: by opening up a gap in the circuit.

The definition of electric current we have developed has the great

virtue that it is easy to measure. In practical electrical work, one

almost always measures current, not charge. The instrument used

to measure current is called an ammeter. A simplified ammeter, g/4,

simply consists of a coiled-wire magnet whose force twists an iron

needle against the resistance of a spring. The greater the current,

the greater the force. Although the construction of ammeters may

differ, their use is always the same. We break into the path of the

electric current and interpose the meter like a tollbooth on a road,

g/5. There is still a complete circuit, and as far as the battery and

bulb are concerned, the ammeter is just another segment of wire.

Does it matter where in the circuit we place the ammeter? Could

we, for instance, have put it in the left side of the circuit instead

of the right? Conservation of charge tells us that this can make no

difference. Charge is not destroyed or “used up” by the lightbulb,

so we will get the same current reading on either side of it. What is

“used up” is energy stored in the battery, which is being converted

into heat and light energy.

5.4 Voltage

The volt unit

Electrical circuits can be used for sending signals, storing infor-

mation, or doing calculations, but their most common purpose by

far is to manipulate energy, as in the battery-and-bulb example of

the previous section. We know that lightbulbs are rated in units of

watts, i.e., how many joules per second of energy they can convert

into heat and light, but how would this relate to the flow of charge as

measured in amperes? By way of analogy, suppose your friend, who

didn’t take physics, can’t find any job better than pitching bales of

hay. The number of calories he burns per hour will certainly depend

on how many bales he pitches per minute, but it will also be pro-

portional to how much mechanical work he has to do on each bale.

If his job is to toss them up into a hayloft, he will got tired a lot

more quickly than someone who merely tips bales off a loading dock

Section 5.4

Voltage

101

into trucks. In metric units,

joules

haybales

joules

=

×

.

second

second

haybale

Similarly, the rate of energy transformation by a battery will not

just depend on how many coulombs per second it pushes through a

circuit but also on how much mechanical work it has to do on each

coulomb of charge:

joules

coulombs

joules

=

×

second

second

coulomb

or

power = current × work per unit charge.

Units of joules per coulomb are abbreviated as volts, 1 V=1 J/C,

named after the Italian physicist Alessandro Volta. Everyone knows

h

that batteries are rated in units of volts, but the voltage concept is

/ Alessandro Volta (1745-1827).

more general than that; it turns out that voltage is a property of

every point in space.

To gain more insight, let’s think again about the analogy with

the haybales. It took a certain number of joules of gravitational

energy to lift a haybale from one level to another. Since we’re talking

about gravitational energy, it really makes more sense to talk about

units of mass, rather than using the haybale as our measure of the

quantity of matter. The gravitational version of voltage would then

be joules per kilogram. Gravitational energy equals mgh, but if we

calculate how much of that we have per kilogram, we’re canceling out

the m, giving simply gh. For any point in the Earth’s gravitational

field, we can assign a number, gh, which tells us how hard it is to

get a given amount of mass to that point. For instance, the top

of Mount Everest would have a big value of gh, because of the big

height. That tells us that it’s expensive in terms of energy to lift a

given amount of mass from some reference level (sea level, say) to

the top of Mount Everest.

Voltage does the same thing, but using electrical energy. We

can visualize an electrical circuit as being like a roller-coaster. The

battery is like the part of the roller-coaster where they lift you up to

the top. The height of this initial hill is analogous to the voltage of

the battery. When you roll downhill later, that’s like a lightbulb. In

the roller-coaster, the initial gravitational energy is turned into heat

and sound as the cars go down the hill. In our circuit, the initial

electrical energy is turned into heat by the lightbulb (and the hot

filament of the lightbulb then glows, turning the heat into light).

Energy stored in a battery

example 3

. The 1.2 V rechargeable battery in figure i is labeled 1800 milliamp-

i / Example 3.

hours. What is the maximum amount of energy the battery can

store?

102

Chapter 5

Electricity

. An ampere-hour is a unit of current multiplied by a unit of time.

Current is charge per unit time, so an ampere-hour is in fact a

funny unit of charge:

(1 A)(1 hour) = (1 C/s)(3600 s)

= 3600 C

1800 milliamp-hours is therefore 1800 × 10−3 × 3600 C = 6.5 ×

103 C. That’s a huge number of charged particles, but the total

loss of electrical energy will just be their total charge multiplied by

the voltage difference across which they move:

energy = (6.5 × 103 C)(1.2 V)

= 7.8 kJ

Using the definition of voltage, V , we can rewrite the equation

power = current × work per unit charge more concisely as P = IV .

Units of volt-amps

example 4

. Doorbells are often rated in volt-amps. What does this combi-

nation of units mean?

. Current times voltage gives units of power, P = IV , so volt-amps

are really just a nonstandard way of writing watts. They are telling

you how much power the doorbell requires.

Power dissipated by a battery and bulb

example 5

. If a 9.0-volt battery causes 1.0 A to flow through a lightbulb, how

much power is dissipated?

. The voltage rating of a battery tells us what voltage difference

∆V it is designed to maintain between its terminals.

P = I ∆V

= 9.0 A · V

C

J

= 9.0

·

s

C

= 9.0 J/s

= 9.0 W

The only nontrivial thing in this problem was dealing with the units.

One quickly gets used to translating common combinations like

A · V into simpler terms.

Discussion questions

A

In the roller-coaster metaphor, what would a high-voltage roller

coaster be like? What would a high-current roller coaster be like?

B

Criticize the following statements:

“He touched the wire, and 10000 volts went through him.”

Section 5.4

Voltage

103

“That battery has a charge of 9 volts.”

“You used up the charge of the battery.”

C

When you touch a 9-volt battery to your tongue, both positive and

negative ions move through your saliva. Which ions go which way?

D

I once touched a piece of physics apparatus that had been wired

incorrectly, and got a several-thousand-volt voltage difference across my

hand. I was not injured. For what possible reason would the shock have

had insufficient power to hurt me?

5.5 Resistance

What’s the physical difference between a 100-watt lightbulb and a

200-watt one? They both plug into a 110-volt outlet, so according

to the equation P = IV , the only way to explain the double power

of the 200-watt bulb is that it must pull in, or “draw,” twice as

much current. By analogy, a fire hose and a garden hose might be

served by pumps that give the same pressure (voltage), but more

water will flow through the fire hose, because there’s simply more

water in the hose that can flow. Likewise, a wide, deep river could

flow down the same slope as a tiny creek, but the number of liters

of water flowing through the big river is greater. If you look at the

filaments of a 100-watt bulb and a 200-watt bulb, you’ll see that the

200-watt bulb’s filament is thicker. In the charged-particle model

of electricity, we expect that the thicker filament will contain more

charged particles that are available to flow. We say that the thicker

filament has a lower electrical resistance than the thinner one.

k / Georg

Simon

Ohm

(1787-

1854).

j / A fat pipe has less resistance

than a skinny pipe.

Although it’s harder to pump water rapidly through a garden

hose than through a fire hose, we could always compensate by using

a higher-pressure pump. Similarly, the amount of current that will

flow through a lightbulb depends not just on its resistance but also

on how much of a voltage difference is applied across it. For many

substances, including the tungsten metal that lightbulb filaments

are made of, we find that the amount of current that flows is pro-

104

Chapter 5

Electricity

portional to the voltage difference applied to it, so that the ratio of voltage to current stays the same. We then use this ratio as a

numerical definition of resistance,

V

R =

,

I

which is known as Ohm’s law. The units of resistance are ohms,

symbolized with an uppercase Greek letter Omega, Ω. Physically,

when a current flows through a resistance, the result is to transform

electrical energy into heat. In a lightbulb filament, for example, the

heat is what causes the bulb to glow.

Ohm’s law states that many substances, including many solids

and some liquids, display this kind of behavior, at least for voltages

that are not too large. The fact that Ohm’s law is called a “law”

should not be taken to mean that all materials obey it, or that it

has the same fundamental importance as the conservation laws, for

example. Materials are called ohmic or nonohmic, depending on

whether they obey Ohm’s law.

On an intuitive level, we can understand the idea of resistance

by making the sounds “hhhhhh” and “ffffff.” To make air flow out

of your mouth, you use your diaphragm to compress the air in your

chest. The pressure difference between your chest and the air out-

side your mouth is analogous to a voltage difference. When you

make the “h” sound, you form your mouth and throat in a way that

allows air to flow easily. The large flow of air is like a large current.

Dividing by a large current in the definition of resistance means that

we get a small resistance. We say that the small resistance of your

mouth and throat allows a large current to flow. When you make

the “f” sound, you increase the resistance and cause a smaller cur-

rent to flow. In this mechanical analogy, resistance is like friction:

the air rubs against your lips. Mechanical friction converts mechan-

ical forms of energy to heat, as when you rub your hands together.

Electrical friction — resistance — converts electrical energy to heat.

If objects of the same size and shape made from two different

ohmic materials have different resistances, we can say that one ma-

terial is more resistive than the other, or equivalently that it is less

conductive. Materials, such as metals, that are very conductive are

said to be good conductors. Those that are extremely poor conduc-

tors, for example wood or rubber, are classified as insulators. There

is no sharp distinction between the two classes of materials. Some,

such as silicon, lie midway between the two extremes, and are called

semiconductors.

Applications

Superconductors

All materials display some variation in resistance according to

temperature (a fact that is used in thermostats to make a ther-

Section 5.5

Resistance

105

mometer that can be easily interfaced to an electric circuit). More spectacularly, most metals have been found to exhibit a sudden

change to zero resistance when cooled to a certain critical tempera-

ture. They are then said to be superconductors. A current flowing

through a superconductor doesn’t create any heat at all.

Theoretically, superconductors should make a great many excit-

ing devices possible, for example coiled-wire magnets that could be

used to levitate trains. In practice, the critical temperatures of all

metals are very low, and the resulting need for extreme refrigera-

tion has made their use uneconomical except for such specialized

applications as particle accelerators for physics research.

But scientists have recently made the surprising discovery that

certain ceramics are superconductors at less extreme temperatures.

The technological barrier is now in finding practical methods for

making wire out of these brittle materials. Wall Street is currently

investing billions of dollars in developing superconducting devices

for cellular phone relay stations based on these materials. In 2001,

the city of Copenhagen replaced a short section of its electrical power

trunks with superconducing cables, and they are now in operation

and supplying power to customers.

There is currently no satisfactory theory of superconductivity in

general, although superconductivity in metals is understood fairly

well. Unfortunately I have yet to find a fundamental explanation of

superconductivity in metals that works at the introductory level.

Constant voltage throughout a conductor

The idea of a superconductor leads us to the question of how

we should expect an object to behave if it is made of a very good

conductor. Superconductors are an extreme case, but often a metal

wire can be thought of as a perfect conductor, for example if the

parts of the circuit other than the wire are made of much less con-

ductive materials. What happens if the resistance equals zero in the

equation

V

R =

?

I

The result of dividing two numbers can only be zero if the number

on top equals zero. This tells us that if we pick any two points

in a perfect conductor, the voltage difference between them must

be zero. In other words, the entire conductor must be at the same

voltage. Using the water metaphor, a perfect conductor is like a

perfectly calm lake or canal, whose surface is flat. If you take an

eyedropper and deposit a drop of water anywhere on the surface, it

doesn’t flow away, because the water is still. In electrical terms, a

charge located anywhere in the interior of a perfect conductor will

always feel a total electrical force of zero.

Suppose, for example, that you build up a static charge by scuff-

106

Chapter 5

Electricity

ing your feet on a carpet, and then you deposit some of that charge

onto a doorknob, which is a good conductor. How can all that charge

be in the doorknob without creating any electrical force at any point

inside it? The only possible answer is that the charge moves around

until it has spread itself into just the right configuration. In this

configuration, the forces exerted by all the charge on any charged

particle within the doorknob exactly cancel out.

We can explain this behavior if we assume that the charge placed

on the doorknob eventually settles down into a stable equilibrium.

Since the doorknob is a conductor, the charge is free to move through

it. If it was free to move and any part of it did experience a nonzero

total force from the rest of the charge, then it would move, and we

would not have an equilibrium.

It also turns out that charge placed on a conductor, once it

reaches its equilibrium configuration, is entirely on the surface, not

on the interior. We will not prove this fact formally, but it is intu-

itively reasonable (see discussion question B).

Short circuits

So far we have been assuming a perfect conductor. What if it’s

a good conductor, but not a perfect one? Then we can solve for

V = IR.

An ordinary-sized current will make a very small result when we

multiply it by the resistance of a good conductor such as a metal

wire. The voltage throughout the wire will then be nearly constant.

If, on the other hand, the current is extremely large, we can have

a significant voltage difference. This is what happens in a short-

circuit: a circuit in which a low-resistance pathway connects the two

sides of a voltage source. Note that this is much more specific than

the popular use of the term to indicate any electrical malfunction

at all. If, for example, you short-circuit a 9-volt battery as shown in

the figure, you will produce perhaps a thousand amperes of current,

leading to a very large value of P = IV . The wire gets hot!

The voltmeter

A voltmeter is nothing more than an ammeter with an additional

high-value resistor through which the current is also forced to flow,

l/1.

Ohm’s law relates the current through the resistor directly

to the voltage difference across it, so the meter can be calibrated

in units of volts based on the known value of the resistor. The

l / 1.

A simplified diagram of

how a voltmeter works. 2. Mea-

voltmeter’s two probes are touched to the two locations in a circuit

suring

the

voltage

difference

between which we wish to measure the voltage difference, l/2. Note

across a lightbulb. 3. The same

how cumbersome this type of drawing is, and how difficult it can

setup drawn in schematic form. 4.

be to tell what is connected to what. This is why electrical drawing

The setup for measuring current

are usually shown in schematic form.

Figure l/3 is a schematic

is different.

representation of figure l/2.

Section 5.5

Resistance

107

The setups for measuring current and voltage are different. When

we’re measuring current, we’re finding “how much stuff goes through,”

so we place the ammeter where all the current is forced to go through

it. Voltage, however, is not “stuff that goes through,” it is a measure

of electrical energy. If an ammeter is like the meter that measures

your water use, a voltmeter is like a measuring stick that tells you

how high a waterfall is, so that you can determine how much energy

will be released by each kilogram of falling water. We don’t want to

force the water to go through the measuring stick! The arrangement

in figure l/3 is a parallel circuit: one which in there are “forks in

the road” where some of the current will flow one way and some

will flow the other. Figure l/4 is said to be wired in series: all the

current will visit all the circuit elements one after the other.

If you inserted a voltmeter incorrectly, in series with the bulb and

battery, its large internal resistance would cut the current down so

low that the bulb would go out. You would have severely disturbed

the behavior of the circuit by trying to measure something about it.

Incorrectly placing an ammeter in parallel is likely to be even

more disconcerting. The ammeter has nothing but wire inside it to

provide resistance, so given the choice, most of the current will flow

through it rather than through the bulb. So much current will flow

through the ammeter, in fact, that there is a danger of burning out

the battery or the meter or both! For this reason, most ammeters

have fuses or circuit breakers inside. Some models will trip their

circuit breakers and make an audible alarm in this situation, while

others will simply blow a fuse and stop working until you replace it.

Discussion questions

A

In figure g/4 on page 100, what would happen if you had the ammeter

on the left rather than on the right?

B

Imagine a charged doorknob, as described on page 107. Why is

it intuitively reasonable to believe that all the charge will end up on the

surface of the doorknob, rather than on the interior?

108

Chapter 5

Electricity

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

A hydrogen atom consists of an electron and a proton. For our

present purposes, we’ll think of the electron as orbiting in a circle

around the proton.

The subatomic particles called muons behave exactly like electrons,

except that a muon’s mass is greater by a factor of 206.77. Muons

are continually bombarding the Earth as part of the stream of par-

ticles from space known as cosmic rays. When a muon strikes an

atom, it can displace one of its electrons. If the atom happens to be

a hydrogen atom, then the muon takes up an orbit that is on the av-

erage 206.77 times closer to the proton than the orbit of the ejected

electron. How many times greater is the electric force experienced

by the muon than that previously felt by the electron?

2

The figure shows a circuit containing five lightbulbs con-

nected to a battery. Suppose you’re going to connect one probe of a

voltmeter to the circuit at the point marked with a dot. How many

unique, nonzero voltage differences could you measure by connecting

the other probe to other wires in the circuit? Visualize the circuit

using the same waterfall metaphor.

3

The lightbulbs in the figure are all identical. If you were

inserting an ammeter at various places in the circuit, how many

unique currents could you measure? If you know that the current

measurement will give the same number in more than one place,

only count that as one unique current.

Problems 2 and 3.

Problems

109

110

Chapter 5

Electricity

This sunspot is a product of the sun’s magnetic fields. The darkest region

in the center is about the size of our planet.

Chapter 6

Fields

6.1 Farewell to the mechanical universe

As late as 1900, physicists generally conceived of the universe in

mechanical terms. Newton had revealed the solar system as a col-

lection of material objects interacting through forces that acted at

a distance. By 1900, evidence began to accumulate for the exis-

tence of atoms as real things, and not just as imaginary models

of reality. In this microscopic realm, the same (successful) Newto-

nian picture tended to be transferred over to the microscopic world.

Now the actors on the stage were atoms rather than planets, and

the forces were electrical rather than gravitational, but it seemed to

be a variation on the same theme. Some physicists, however, began

to realize that the old mechanical picture wouldn’t quite work. At

a deeper level, the operation of the universe came to be understood

in terms of fields, the general idea being embodied fairly well in

“The Force” from the Star Wars movies: “... an energy field cre-

ated by all living things. It surrounds us, penetrates us, and binds

the galaxy together.” Substitute “massive” for “living,” and you

have a fairly good description of the gravitational field, which I first

casually mentioned on page 20. Substitute “charged” instead, and

111

it’s a depiction of the electric field.

Time delays in forces exerted at a distance

What convinced physicists that they needed this new concept of

a field of force? Although we have been dealing mostly with elec-

trical forces, let’s start with a magnetic example. (In fact the main

reason I’ve delayed a detailed discussion of magnetism for so long

is that mathematical calculations of magnetic effects are handled

much more easily with the concept of a field of force.) First a little

background leading up to our example. A bar magnet, a, has an axis

about which many of the electrons’ orbits are oriented. The earth

itself is also a magnet, although not a bar-shaped one. The interac-

tion between the earth-magnet and the bar magnet, b, makes them

want to line up their axes in opposing directions (in other words

such that their electrons rotate in parallel planes, but with one set

rotating clockwise and the other counterclockwise as seen looking

along the axes). On a smaller scale, any two bar magnets placed

near each other will try to align themselves head-to-tail, c.

Now we get to the relevant example. It is clear that two people

separated by a paper-thin wall could use a pair of bar magnets to

signal to each other. Each person would feel her own magnet trying

to twist around in response to any rotation performed by the other

person’s magnet. The practical range of communication would be

very short for this setup, but a sensitive electrical apparatus could

a / A bar magnet’s atoms are

pick up magnetic signals from much farther away. In fact, this is

(partially) aligned.

not so different from what a radio does: the electrons racing up

and down the transmitting antenna create forces on the electrons

in the distant receiving antenna. (Both magnetic and electric forces

are involved in real radio signals, but we don’t need to worry about

that yet.)

A question now naturally arises as to whether there is any time

delay in this kind of communication via magnetic (and electric)

forces.

Newton would have thought not, since he conceived of

physics in terms of instantaneous action at a distance. We now

know, however, that there is such a time delay.

If you make a

long-distance phone call that is routed through a communications

satellite, you should easily be able to detect a delay of about half a

b / A

bar

magnet

interacts

second over the signal’s round trip of 50,000 miles. Modern measure-

with our magnetic planet.

ments have shown that electric, magnetic, and gravitational forces

all travel at the speed of light, 3 × 108 m/s.1 (In fact, we will soon

discuss how light itself is made of electricity and magnetism.)

If it takes some time for forces to be transmitted through space,

then apparently there is some thing that travels through space. The

fact that the phenomenon travels outward at the same speed in all

1As discussed in chapter 4, one consequence of Einstein’s theory of relativity

c / Magnets aligned north-south.

is that material objects can never move faster than the speed of light. It can

also be shown that signals or information are subject to the same limit.

112

Chapter 6

Fields

directions strongly evokes wave metaphors such as ripples on a pond.

More evidence that fields of force are real: they carry energy.

The smoking-gun argument for this strange notion of traveling

force ripples comes from the fact that they carry energy.

First suppose that the person holding the bar magnet on the

right decides to reverse hers, resulting in configuration d. She had

to do mechanical work to twist it, and if she releases the magnet,

d / The second magnet is re-

versed.

energy will be released as it flips back to c. She has apparently stored

energy by going from c to d. So far everything is easily explained

without the concept of a field of force.

But now imagine that the two people start in position c and

then simultaneously flip their magnets extremely quickly to position

e, keeping them lined up with each other the whole time. Imagine,

for the sake of argument, that they can do this so quickly that

each magnet is reversed while the force signal from the other is

e / Both magnets are reversed.

still in transit. (For a more realistic example, we’d have to have

two radio antennas, not two magnets, but the magnets are easier

to visualize.) During the flipping, each magnet is still feeling the

forces arising from the way the other magnet used to be oriented.

Even though the two magnets stay aligned during the flip, the time

delay causes each person to feel resistance as she twists her magnet

around.

How can this be?

Both of them are apparently doing

mechanical work, so they must be storing magnetic energy somehow.

But in the traditional Newtonian conception of matter interacting

via instantaneous forces at a distance, interaction energy arises from

the relative positions of objects that are interacting via forces. If

the magnets never changed their orientations relative to each other,

how can any magnetic energy have been stored?

The only possible answer is that the energy must have gone

into the magnetic force ripples crisscrossing the space between the

magnets. Fields of force apparently carry energy across space, which

is strong evidence that they are real things.

This is perhaps not as radical an idea to us as it was to our

ancestors. We are used to the idea that a radio transmitting antenna

consumes a great deal of power, and somehow spews it out into the

universe. A person working around such an antenna needs to be

careful not to get too close to it, since all that energy can easily

cook flesh (a painful phenomenon known as an “RF burn”).

The gravitational field

f / The

wind

patterns

in

a

certain area of the ocean could

Given that fields of force are real, how do we define, measure,

be charted in a “sea of arrows”

and calculate them? A fruitful metaphor will be the wind patterns

representation like this.

Each

experienced by a sailing ship. Wherever the ship goes, it will feel a

arrow represents both the wind’s

strength and its direction at a

certain amount of force from the wind, and that force will be in a

certain location.

certain direction. The weather is ever-changing, of course, but for

Section 6.1

Farewell to the mechanical universe

113

now let’s just imagine steady wind patterns. Definitions in physics

are operational, i.e., they describe how to measure the thing being

defined. The ship’s captain can measure the wind’s “field of force”

by going to the location of interest and determining both the direc-

tion of the wind and the strength with which it is blowing. Charting

all these measurements on a map leads to a depiction of the field of

wind force like the one shown in the figure. This is known as the

“sea of arrows” method of visualizing a field.

Now let’s see how these concepts are applied to the fundamental

force fields of the universe. We’ll start with the gravitational field,

which is the easiest to understand. We’ve already encountered the

gravitational field, g, which we defined in terms of energy. Essen-

tially, g was defined as the number that would make the equation

GE = mgh give the right answer. However, we intuitively feel that

the gravitational field has a direction associated with it: down! This

can be more easily expressed via the following definition:

definition of the gravitational field

The gravitational field, g, at any location in space is found by

placing a test mass m at that point. The field is then given

by g = F/m, where F is the gravitational force on the test

mass.

With this new definition, we get units of N/kg, rather then

J/kg/m. These are in fact equivalent units.

The most subtle point about all this is that the gravitational

field tells us about what forces would be exerted on a test mass by

the earth, sun, moon, and the rest of the universe, if we inserted a

test mass at the point in question. The field still exists at all the

places where we didn’t measure it.

Sources and sinks

If we make a sea-of-arrows picture of the gravitational fields

surrounding the earth, g, the result is evocative of water going down

a drain. For this reason, anything that creates an inward-pointing

field around itself is called a sink. The earth is a gravitational sink.

The term “source” can refer specifically to things that make outward

fields, or it can be used as a more general term for both “outies”

and “innies.” However confusing the terminology, we know that

gravitational fields are only attractive, so we will never find a region

of space with an outward-pointing field pattern.

Knowledge of the field is interchangeable with knowledge of its

g / The

gravitational

field

sur-

sources (at least in the case of a static, unchanging field). If aliens

rounding a clump of mass such

saw the earth’s gravitational field pattern they could immediately

as the earth.

infer the existence of the planet, and conversely if they knew the

mass of the earth they could predict its influence on the surrounding

gravitational field.

114

Chapter 6

Fields

The electric field

The definition of the electric field is directly analogous to, and

has the same motivation as, the definition of the gravitational field:

definition of the electric field

The electric field, E, at any location in space is found by

placing a test charge q at that point. The electric field vector

is then given by E = F/q, where F is the electric force on the

test charge.

Charges are what create electric fields. Unlike gravity, which is

always attractive, electricity displays both attraction and repulsion.

A positive charge is a source of electric fields, and a negative one is

a sink.

6.2 Electromagnetism

Think not that I am come to destroy the law, or the prophets: I am

not come to destroy, but to fulfill.

Matthew 5:17

Magnetic interactions

At this stage, you understand roughly as much about the clas-

sification of interactions as physicists understood around the year

1800. There appear to be three fundamentally different types of

interactions: gravitational, electrical, and magnetic. Many types

of interactions that appear superficially to be distinct — stickiness,

chemical interactions, the energy an archer stores in a bow — are

really the same: they’re manifestations of electrical interactions be-

tween atoms. Is there any way to shorten the list any further? The

prospects seem dim at first. For instance, we find that if we rub a

piece of fur on a rubber rod, the fur does not attract or repel a mag-

net. The fur has an electric field, and the magnet has a magnetic

field. The two are completely separate, and don’t seem to affect

one another. Likewise we can test whether magnetizing a piece of

iron changes its weight. The weight doesn’t seem to change by any

measurable amount, so magnetism and gravity seem to be unrelated.

h / 1.

When the circuit is in-

complete,

no

current

flows

That was where things stood until 1820, when the Danish physi-

through the wire, and the magnet

cist Hans Christian Oersted was delivering a lecture at the Univer-

is unaffected.

It points in the

sity of Copenhagen, and he wanted to give his students a demonstra-

direction of the Earth’s magnetic

tion that would illustrate the cutting edge of research. He generated

field. 2. The circuit is completed,

a current in a wire by making a short circuit across a battery, and

and current flows through the

wire.

The wire has a strong

held the wire near a magnetic compass. The ideas was to give an

effect on the magnet, which turns

example of how one could search for a previously undiscovered link

almost perpendicular to it. If the

between electricity (the electric current in the wire) and magnetism.

earth’s field could be removed

One never knows how much to believe from these dramatic legends,

entirely, the compass would point

but the story is2 that the experiment he’d expected to turn out neg-

exactly perpendicular to the wire;

this is the direction of the wire’s

2Oersted’s paper describing the phenomenon says that “The first experiments

field.

on the subject . . . were set on foot in the classes for electricity, galvanism, and Section 6.2

Electromagnetism

115

ative instead turned out positive: when he held the wire near the

compass, the current in the wire caused the compass to twist!

People had tried similar experiments before, but only with static

electricity, not with a moving electric current. For instance, they had

hung batteries so that they were free to rotate in the earth’s mag-

netic field, and found no effect; since the battery was not connected

to a complete circuit, there was no current flowing. With Oersted’s

own setup, h, the effect was only produced when the “circuit was

closed, but not when open, as certain very celebrated physicists in

vain attempted several years ago.”3

Oersted was eventually led to the conclusion that magnetism was

an interaction between moving charges and other moving charges,

i.e., between one current and another. A permanent magnet, he in-

ferred, contained currents on a microscopic scale that simply weren’t

i / A

schematic

representa-

tion of an unmagnetized material,

practical to measure with an ammeter. Today this seems natural

1, and a magnetized one, 2.

to us, since we’re accustomed to picturing an atom as a tiny solar

system, with the electrons whizzing around the nucleus in circles.

As shown in figure i, a magnetized piece of iron is different from an

unmagnetized piece because the atoms in the unmagnetized piece

are jumbled in random orientations, whereas the atoms in the mag-

netized piece are at least partially organized to face in a certain

direction.

Figure j shows an example that is conceptually simple, but not

very practical. If you try this with a typical vacuum tube, like a

TV or computer monitor, the current in the wire probably won’t be

enough to produce a visible effect. A more practical method is to

hold a magnet near the screen. We still have an interaction between

moving charges and moving charges, but the swirling electrons in

the atoms in the magnet are now playing the role played by the

moving charges in the wire in figure j. Warning: if you do this,

j / Magnetism

is

an

interac-

make sure your monitor has a demagnetizing button! If not, then

tion between moving charges

your monitor may be permanently ruined.

and moving charges. The moving

charges in the wire attract the

Relativity requires magnetism

moving charges in the beam of

So magnetism is an interaction between moving charges and

charged particles in the vacuum

tube.

moving charges. But how can that be? Relativity tells us that

motion is a matter of opinion. Consider figure k. In this figure and

in figure l, the dark and light coloring of the particles represents the

fact that one particle has one type of charge and the other particle

has the other type. Observer k/2 sees the two particles as flying

through space side by side, so they would interact both electrically

(simply because they’re charged) and magnetically (because they’re

charges in motion). But an observer moving along with them, k/1,

would say they were both at rest, and would expect only an elec-

magnetism, which were held by me in the winter just past,” but that doesn’t tell

us whether the result was really a surprise that occurred in front of his students.

3All quotes are from the 1876 translation by J.E. Kempe.

116

Chapter 6

Fields

trical interaction. This seems like a paradox. Magnetism, however,

comes not to destroy relativity but to fulfill it. Magnetic interac-

tions must exist according to the theory of relativity. To understand

how this can be, consider how time and space behave in relativity.

Observers in different frames of reference disagree about the lengths

of measuring sticks and the speeds of clocks, but the laws of physics

are valid and self-consistent in either frame of reference. Similarly,

observers in different frames of reference disagree about what elec-

tric and magnetic fields there are, but they agree about concrete

physical events. An observer in frame of reference k/1 says there

are electric fields around the particles, and predicts that as time

goes on, the particles will begin to accelerate towards one another,

k / One observer sees an electric

eventually colliding. She explains the collision as being due to the

field, while the other sees both an

electrical attraction between the particles. A different observer, k/2,

electric field and a magnetic one.

says the particles are moving. This observer also predicts that the

particles will collide, but explains their motion in terms of both an

electric field and a magnetic field. As we’ll see shortly, the mag-

netic field is required in order to maintain consistency between the

predictions made in the two frames of reference.

To see how this really works out, we need to find a nice simple

example. An example like figure k is not easy to handle, because in

the second frame of reference, the moving charges create fields that

change over time at any given location, like when the V-shaped wake

of a speedboat washes over a buoy. Examples like figure j are easier,

because there is a steady flow of charges, and all the fields stay the

same over time. Figure l/1 shows a simplified and idealized model

of figure j. The charge by itself is like one of the charged particles in

the vacuum tube beam of figure j, and instead of the wire, we have

two long lines of charges moving in opposite directions. Note that,

l / A model of a charged par-

as discussed in discussion question C on page 104, the currents of

ticle and a current-carrying wire,

the two lines of charges do not cancel out. The dark balls represent

seen in two different frames of

particles with one type of charge, and the light balls have the other

reference. The relativistic length

type. Because of this, the total current in the “wire” is double what

contraction is highly exaggerated.

it would be if we took away one line.

The force on the lone particle is

purely magnetic in 1, and purely

As a model of figure j, figure l/1 is partly realistic and partly

electric in 2.

unrealistic. In a real piece of copper wire, there are indeed charged

particles of both types, but it turns out that the particles of one

type (the protons) are locked in place, while only some of the other

type (the electrons) are free to move. The model also shows the

particles moving in a simple and orderly way, like cars on a two-

lane road, whereas in reality most of the particles are organized

into copper atoms, and there is also a great deal of random thermal

motion. The model’s unrealistic features aren’t a problem, because

the point of this exercise is only to find one particular situation that

shows magnetic effects must exist based on relativity.

What electrical force does the lone particle in figure l/1 feel?

Since the density of “traffic” on the two sides of the “road” is equal,

Section 6.2

Electromagnetism

117

there is zero overall electrical force on the lone particle. Each “car”

that attracts the lone particle is paired with a partner on the other

side of the road that repels it. If we didn’t know about magnetism,

we’d think this was the whole story: the lone particle feels no force

at all from the wire.

Figure l/2 shows what we’d see if we were observing all this from

a frame of reference moving along with the lone charge. Here’s where

the relativity comes in. Relativity tells us that moving objects ap-

pear contracted to an observer who is not moving along with them.

Both lines of charge are in motion in both frames of reference, but

in frame 1 they were moving at equal speeds, so their contractions

were equal. In frame 2, however, their speeds are unequal. The

dark charges are moving more slowly than in frame 1, so in frame 2

they are less contracted. The light-colored charges are moving more

quickly, so their contraction is greater now. The “cars” on the two

sides of the “road” are no longer paired off, so the electrical forces

on the lone particle no longer cancel out as they did in l/1. The

lone particle is attracted to the wire, because the particles attract-

ing it are more dense than the ones repelling it. Furthermore, the

attraction felt by the lone charge must be purely electrical, since the

lone charge is at rest in this frame of reference, and magnetic effects

occur only between moving charges and other moving charges.

Now observers in frames 1 and 2 disagree about many things,

but they do agree on concrete events. Observer 2 is going to see

the lone particle drift toward the wire due to the wire’s electrical

attraction, gradually speeding up, and eventually hit the wire. If 2

sees this collision, then 1 must as well. But 1 knows that the total

electrical force on the lone particle is exactly zero. There must be

some new type of force. She invents a name for this new type of

force: magnetism. This was a particularly simple example, because

the force was purely magnetic in one frame of reference, and purely

electrical in another. In general, an observer in a certain frame

of reference will measure a mixture of electric and magnetic fields,

while an observer in another frame, in motion with respect to the

first, says that the same volume of space contains a different mixture.

We therefore arrive at the conclusion that electric and magnetic

phenomena aren’t separate. They’re different sides of the same coin.

We refer to electric and magnetic interactions collectively as elec-

tromagnetic interactions. Our list of the fundamental interactions

m / Magnetic

interactions

in-

of nature now has two items on it instead of three: gravity and

volving only two particles at a

electromagnetism.

time.

In these figures, unlike

figure l/1, there are electrical

The basic rules for magnetic attractions and repulsions, shown in

forces as well as magnetic ones.

figure m, aren’t quite as simple as the ones for gravity and electric-

The electrical forces are not

ity. Rules m/1 and m/2 follow directly from our previous analysis

shown here.

Don’t memorize

of figure l. Rules 3 and 4 are obtained by flipping the type of charge

these rules!

that the bottom particle has. For instance, rule 3 is like rule 1,

118

Chapter 6

Fields

except that the bottom charge is now the opposite type. This turns

the attraction into a repulsion. (We know that flipping the charge

reverses the interaction, because that’s the way it works for elec-

tric forces, and magnetic forces are just electric forces viewed in a

different frame of reference.)

A magnetic weathervane placed near a current.

example 1

Figure n shows a magnetic weathervane, consisting of two charges

n / Example 1

that spin in circles around the axis of the arrow. (The magnetic

field doesn’t cause them to spin; a motor is needed to get them to

spin in the first place.) Just like the magnetic compass in figure h,

the weathervane’s arrow tends to align itself in the direction per-

pendicular to the wire. This is its preferred orientation because

the charge close to the wire is attracted to the wire, while the

charge far from the wire is repelled by it.

Magnetic fields

How should we define the magnetic field? When two objects at-

tract each other gravitationally, their gravitational energy depends

only on the distance between them, and it seems intuitively reason-

able that we define the gravitational field arrows like a street sign

that says “this way to lower gravitational energy.” The same idea

works fine for the electric field. But what if two charged particles

are interacting magnetically? Their interaction doesn’t just depend

on the distance, but also on their motions.

We need some way to pick out some direction in space, so we

can say, “this is the direction of the magnetic field around here.” A

natural and simple method is to define the magnetic field’s direction

according to the direction a compass points.

Starting from this

definition we can, for example, do experiments to show that the

magnetic field of a current-carrying wire forms a circular pattern, o.

But is this the right definition? Unlike the definitions of the

gravitational and electric fields’ directions, it involves a particular

human-constructed tool. However, compare figure h on page 115

with figure n on page 119. Note that both of these tools line them-

selves up along a line that’s perpendicular to the wire. In fact, no

matter how hard you try, you will never be able to invent any other

o / The

magnetic

field

curls

electromagnetic device that will align itself with any other line. All

around the wire in circles.

At

you can do is make one that points in exactly the opposite direction,

each point in space, the magnetic

but along the same line. For instance, you could use paint to reverse

compass shows the direction of

the colors that label the ends of the magnetic compass needle, or

the field.

you could build a weathervane just like figure n, but spinning like a

left-handed screw instead of a right-handed one. The weathervane

and the compass aren’t even as different as they appear. Figure p

shows their hidden similarities.

Nature is trying to tell us something: there really is something

Section 6.2

Electromagnetism

119

p / 1. The needle of a magnetic

compass is nothing more than a

bar magnet that is free to rotate in

response to the earth’s magnetic

field. 2. A cartoon of the bar mag-

net’s structure at the atomic level.

Each atom is very much like the

weathervane of figure n.

special about the direction the compass points. Defining the direc-

tion of the magnetic field in terms of this particular device isn’t as

arbitrary as it seems. The only arbitrariness is that we could have

built up a whole self-consistent set of definitions that started by

defining the magnetic field as being in the opposite direction.

Head-to-tail alignment of bar magnets

example 2

. If you let two bar magnets like the one in figure p interact, which

way do they want to line up, head-to-head or head-to-tail?

. Each bar magnet contains a huge number of atoms, but that

won’t matter for our result; we can imagine this as an interac-

tion between two individual atoms. For that matter, let’s model

the atoms as weathervanes like the one in figure n. Suppose we

put two such weather vanes side by side, with their arrows both

pointing away from us. From our point of view, they’re both spin-

ning clockwise. As one of the charges in the left-hand weather

vane comes down on the right side, one of the charges in the

right-hand vane comes up on the left side. These two charges

are close together, so their magnetic interaction is very strong at

this moment. Their interaction is repulsive, so this is an unstable

arrangement of the two weathervanes.

On the other hand, suppose the left-hand weathervane is point-

ing away from is, while its partner on the right is pointing toward

us. From our point of view, we see the one on the right spinning

counterclockwise. At the moment when their charges come as

close as possible, they’re both on the way up. Their interaction is

attractive, so this is a stable arrangement.

Translating back from our model to the original question about bar

magnets, we find that bar magnets will tend to align themselves

head-to-tail. This is easily verified by experiment.

If you go back and apply this definition to all the examples we’ve

encountered so far, you’ll find that there’s a general rule: the force on

120

Chapter 6

Fields

q / The force on a charged par-

ticle moving through a magnetic

field is perpendicular to both the

field and its direction of motion.

The relationship is right-handed

for one type of charge, and left-

handed for the other type.

a charged particle moving through a magnetic field is perpendicular

to both the field and its direction of motion. A force perpendicular

to the direction of motion is exactly what is required for circular

motion, so we find that a charged particle in a vacuum will go in

a circle around the magnetic field arrows (or perhaps a corkscrew

pattern, if it also has some motion along the direction of the field).

That means that magnetic fields tend to trap charged particles.

Figure r shows this principle in action. A beam of electrons is

created in a vacuum tube, in which a small amount of hydrogen

gas has been left. A few of the electrons strike hydrogen molecules,

creating light and letting us see the path of the beam. A magnetic

field is produced by passing a current (meter) through the circular

coils of wire in front of and behind the tube. In the bottom figure,

with the magnetic field turned on, the force perpendicular to the

electrons’ direction of motion causes them to move in a circle.

Sunspots

example 3

Sunspots, like the one shown in the photo on page 111, are

places where the sun’s magnetic field is unusually strong. Charged

particles are trapped there for months at a time. This is enough

time for the sunspot to cool down significantly, and it doesn’t get

heated back up because the hotter surrounding material is kept

out by the same magnetic forces.

r / A

beam

of

electrons

cir-

cles around the magnetic field

The aurora and life on earth’s surface

example 4

arrows.

A strong magnetic field seems to be one of the prerequisites for

the existence of life on the surface of a planet. Energetic charged

particles from the sun are trapped by our planet’s magnetic field,

and harmlessly spiral down to the earth’s surface at the poles.

In addition to protecting us, this creates the aurora, or “northern

lights.”

The astronauts who went to the moon were outside of the earth’s

Section 6.2

Electromagnetism

121

protective field for about a week, and suffered significant doses

of radiation during that time. The problem would be much more

serious for astronauts on a voyage to Mars, which would take

at least a couple of years. They would be subjected to intense

radiation while in interplanetary space, and also while on Mars’s

surface, since Mars lacks a strong magnetic field.

Features in one Martian rock have been interpreted by some sci-

entists as fossilized bacteria. If single-celled life evolved on Mars,

s / Faraday on a British ban-

it has presumably been forced to stay below the surface. (Life

knote.

on Earth probably evolved deep in the oceans, and most of the

Earth’s biomass consists of single-celled organisms in the oceans

and deep underground.)

6.3 Induction

We’ve already seen that the electric and magnetic fields are closely

related, since what one observer sees as one type of field, another

observer in a different frame of reference sees as a mixture of both.

The relationship goes even deeper than that, however.

Figure t

shows an example that doesn’t even involve two different frames of

reference. This phenomenon of induced electric fields — fields that

are not due to charges — was a purely experimental accomplishment

by Michael Faraday (1791-1867), the son of a blacksmith who had

to struggle against the rigid class structure of 19th century Eng-

land. Faraday, working in 1831, had only a vague and general idea

that electricity and magnetism were related to each other, based on

Oersted’s demonstration, a decade before, that magnetic fields were

caused by electric currents.

Figure t is a simplified drawing of the experiment, as described

in Faraday’s original paper: “Two hundred and three feet of cop-

per wire . . . were passed round a large block of wood; [another] two

hundred and three feet of similar wire were interposed as a spiral

between the turns of the first, and metallic contact everywhere pre-

vented by twine [insulation]. One of these [coils] was connected with

a galvanometer [voltmeter], and the other with a battery. . . When

the contact was made, there was a sudden and very slight effect at

the galvanometer, and there was also a similar slight effect when

the contact with the battery was broken. But whilst the . . . current

was continuing to pass through the one [coil], no . . . effect . . . upon

the other [coil] could be perceived, although the active power of the

battery was proved to be great, by its heating the whole of its own

coil [through ordinary resistive heating] . . . ”

From Faraday’s notes and publications, it appears that the sit-

t

uation in figure t/3 was a surprise to him, and he probably thought

/ Faraday’s

experiment,

sim-

plified and shown with modern

it would be a surprise to his readers, as well. That’s why he offered

equipment.

evidence that the current was still flowing: to show that the battery

hadn’t just died. The induction effect occurred during the short

122

Chapter 6

Fields

time it took for the black coil’s magnetic field to be established, t/2.

Even more counterintuitively, we get an effect, equally strong but

in the opposite direction, when the circuit is broken, t/4. The effect

occurs only when the magnetic field is changing: either ramping up

or ramping down.

What are we really measuring here with the voltmeter? A volt-

meter is nothing more than a resistor with an attachment for mea-

suring the current through it. A current will not flow through a

resistor unless there is some electric field pushing the electrons, so

we conclude that the changing magnetic field has produced an elec-

tric field in the surrounding space. Since the white wire is not a

perfect conductor, there must be electric fields in it as well. The

remarkable thing about the circuit formed by the white wire is that

as the electrons travel around and around, they are always being

pushed forward by electric fields. That is, the electric field seems to

form a curly pattern, like a whirlpool.

What Faraday observed was an example of the following princi-

ple:

the principle of induction

Any magnetic field that changes over time will create an electric

u / The

geometry

of

induced

field. The induced electric field is perpendicular to the magnetic

fields. The induced field tends to

field, and forms a curly pattern around it.

form a whirlpool pattern around

Any electric field that changes over time will create a magnetic field.

the change in the field producing

The induced magnetic field is perpendicular to the electric field, and

it. The notation ∆ (Greek letter

forms a curly pattern around it.

delta) stands for “change in.”

Note how the induced fields

circulate in opposite directions.

The first part was the one Faraday had seen in his experiment. The

geometrical relationships are illustrated in figure u. In Faraday’s

setup, the magnetic field was pointing along the axis of the coil of

wire, so the induced electric field made a curly pattern that circled

around the circumference of the block.

The generator

example 5

A basic generator, v, consists of a permanent magnet that rotates

within a coil of wire. The magnet is turned by a motor or crank,

(not shown). As it spins, the nearby magnetic field changes. This

changing magnetic field results in an electric field, which has a

curly pattern. This electric field pattern creates a current that

whips around the coils of wire, and we can tap this current to light

the lightbulb.

If the magnet was on a frictionless bearing, could we light the bulb

for free indefinitely, thus violating conservation of energy? No.

It’s hard work to crank the magnet, and that’s where the energy

comes from. If we break the light-bulb circuit, it suddenly gets

v / A generator.

easier to crank the magnet! This is because the current in the

Section 6.3

Induction

123

coil sets up its own magnetic field, and that field exerts a torque

on the magnet. If we stopped cranking, this torque would quickly

make the magnet stop turning.

self-check A

When you’re driving your car, the engine recharges the battery continu-

ously using a device called an alternator, which is really just a generator.

Why can’t you use the alternator to start the engine if your car’s battery

is dead?

. Answer, p. 180

The transformer

example 6

It’s more efficient for the electric company to transmit power over

electrical lines using high voltages and low currents. However,

we don’t want our wall sockets to operate at 10000 volts! For this

reason, the electric company uses a device called a transformer,

w, to convert everything to lower voltages and higher currents

inside your house. The coil on the input side creates a magnetic

field. Transformers work with alternating current (currents that

reverses its direction many times a second), so the magnetic field

surrounding the input coil is always changing. This induces an

w

electric field, which drives a current around the output coil.

/ A transformer.

Since the electric field is curly, an electron can keep gaining more

and more energy by circling through it again and again. Thus

the output voltage can be controlled by changing the number of

turns of wire on the output side. In any case, conservation of

energy guarantees that the amount of power on the output side

must equal the amount put in originally,

(input current) × (input voltage) = (output current) × (output voltage)

so no matter what factor the voltage is reduced by, the current

is increased by the same factor. This is analogous to a lever. A

crowbar allows you to lift a heavy boulder, but to move the boulder

a centimeter, you may have to move your end of the lever a meter.

The advantage in force comes with a disadvantage in distance.

It’s as though you were allowed to lift a small weight through a

large height rather than a large weight through a small height.

Either way, the energy you expend is the same.

Fun with sparks

example 7

Unplug a lamp while it’s turned on, and watch the area around

the wall outlet. You should see a blue spark in the air at the mo-

x / Observer

A

sees

a

posi-

ment when the prongs of the plug lose contact with the electrical

tively charged particle moves

through

a

region

of

upward

contacts inside the socket.

magnetic field, which we assume

This is evidence that, as discussed on page 113, fields contain

to be uniform, between the poles

energy. Somewhere on your street is a transformer, one side

of two magnets.

The resulting

of which is connected to the lamp’s circuit. When the lamp is

force along the z axis causes the

particle’s path to curve toward us.

plugged in and turned on, there’s a complete circuit, and current

flows. As current flows through the coils in the transformer, a

magnetic field is formed — remember, any time there’s moving

124

Chapter 6

Fields

charge, there will be magnetic fields. Because there is a large

number turns in the coils, these fields are fairly strong, and store

quite a bit of energy.

When you pull the plug, the circuit is no longer complete, and the

current stops. Once the current has disappeared, there’s no more

magnetic field, which means that some energy has disappeared.

Conservation of energy tells us that if a certain amount of energy

disappears, an equal amount must reappear somewhere else.

That energy goes into making the spark. (Once the spark is gone,

its energy remains in the form of heat in the air.)

We now have two connections between electric and magnetic

fields. One is the principle of induction, and the other is the idea

that according to relativity, observers in different frames of reference

must perceive different mixtures of magnetic and electric fields. At

the time Faraday was working, relativity was still 70 years in the

future, so the relativistic concepts weren’t available — to him, his

observations were just surprising empirical facts. But in fact, the

relativistic idea about frames of reference has a logical connection

to the idea of induction.

Figure x is a nice example that can be interpreted either way.

Observer A is at rest with respect to the bar magnets, and sees

the particle swerving off in the z direction, as it should according

to the right-hand rule. Suppose observer B, on the other hand, is

moving to the right along the x axis, initially at the same speed

as the particle. B sees the bar magnets moving to the left and the

particle initially at rest but then accelerating along the z axis in a

straight line. It is not possible for a magnetic field to start a particle

moving if it is initially at rest, since magnetism is an interaction of

moving charges with moving charges. B is thus led to the inescapable

conclusion that there is an electric field in this region of space, which

points along the z axis. In other words, what A perceives as a pure

magnetic field, B sees as a mixture of electric and magnetic fields.

This is what we expect based on the relativistic arguments, but it’s

also what’s required by the principle of induction. In B’s frame of

reference, there’s initially no magnetic field, but then a couple of

bar magnets come barging in and create one. This is a change in

the magnetic field, so the principle of induction predicts that there

must be an electric field as well.

Electromagnetic waves

Theorist James Clerk Maxwell was the first to work out the prin-

ciple of induction (including the detailed numerical and geometric

relationships, which we won’t go into here). Legend has it that it

was on a starry night that he first realized the most important im-

plication of his equations: light itself is an electromagnetic wave,

y / James Clerk Maxwell (1831-

a ripple spreading outward from a disturbance in the electric and

1879)

magnetic fields. He went for a walk with his wife, and told her

Section 6.3

Induction

125

she was the only other person in the world who really knew what

starlight was.

The principle of induction tells us that there can be no such

thing as a purely electric or purely magnetic wave. As an electric

wave washes over you, you feel an electric field that changes over

time. By the principle of induction, there must also be a magnetic

field accompanying it. It works the other way, too. It may seem a

little spooky that the electric field causes the magnetic field while

the magnetic field causes the electric field, but the waves themselves

don’t seem to worry about it.

The distance from one ripple to the next is called the wavelength

of the light. Light with a certain wavelength (about quarter a mil-

lionth of a meter) is at the violet end of the rainbow spectrum, while

light with a somewhat longer wavelength (about twice as long) is red.

Figure z/1 shows the complete spectrum of light waves. Maxwell’s

equations predict that all light waves have the same structure, re-

gardless of wavelength and frequency, so even though radio and x-

rays, for example, hadn’t been discovered, Maxwell predicted that

such waves would have to exist. Maxwell’s 1865 prediction passed an

important test in 1888, when Heinrich Hertz published the results

of experiments in which he showed that radio waves could be ma-

nipulated in the same ways as visible light waves. Hertz showed, for

example, that radio waves could be reflected from a flat surface, and

that the directions of the reflected and incoming waves were related

in the same way as with light waves, forming equal angles with the

normal. Likewise, light waves can be focused with a curved, dish-

shaped mirror, and Hertz demonstrated the same thing with radio

waves using a metal dish.

126

Chapter 6

Fields

z / Panel 1 shows the electro-

magnetic spectrum.

Panel

2 shows how an electromagnetic

wave is put together. Imagine that

this is a radio wave, with a wave-

length of a few meters.

If you

were standing inside the wave

as it passed through you, you

could theoretically hold a com-

pass in your hand, and it would

wiggle back and forth as the mag-

netic field pattern (white arrows)

washed over you.

(The vibra-

tion would actually be much to

rapid to detect this way.)

Sim-

ilarly, you’d experience an elec-

tric field alternating between up

and down.

Panel 3 shows

how this relates to the principle of

induction. The changing electric

field (black arrows) should create

a curly magnetic field (white). Is it

really curly? Yes, because if we

inserted a paddlewheel that re-

sponded to electric fields, the field

would make the paddlewheel spin

counterclockwise as seen from

above.

Similarly, the changing

magnetic field (white) makes an

electric field (black) that curls in

the clockwise direction as seen

from the front.

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

Albert Einstein wrote, “What really interests me is whether

God had any choice in the creation of the world.” What he meant by

this is that if you randomly try to imagine a set of rules — the laws of

physics — by which the universe works, you’ll almost certainly come

up with rules that don’t make sense. For instance, we’ve seen that

if you tried to omit magnetism from the laws of physics, electrical

interactions wouldn’t make sense as seen by observers in different

frames of reference; magnetism is required by relativity.

The magnetic interaction rules in figure m are consistent with the

time-reversal symmetry of the laws of physics. In other words, the

rules still work correctly if you reverse the particles’ directions of

motion. Now you get to play God (and fail). Suppose you’re going

to make an alternative version of the laws of physics by reversing

Problems

127

the direction of motion of only one of the eight particles. You have eight choices, and each of these eight choices would result in a new

set of physical laws. We can imagine eight alternate universes, each

governed by one of these eight sets. Prove that all of these mod-

ified sets of physical laws are impossible, either because the are

self-contradictory, or because they violate time-reversal symmetry.

2

The purpose of this problem is to show that the magnetic in-

teraction rules shown in figure m can be simplified by stating them

in terms of current. Recall that, as discussed in discussion question

C on page 104, one type of charge moving in a particular direction

produces the same current as the other type of charge moving in

the opposite direction. Let’s say arbitrarily that the current made

by the dark type of charged particle is in the direction it’s moving,

while a light-colored particle produces a current in the direction op-

posite to its motion. Redraw all four panels of figure m, replacing

each picture of a moving light or dark particle with an arrow show-

ing the direction of the current it makes. Show that the rules for

attraction and repulsion can now be made much simpler, and state

the simplified rules explicitly.

3

Physicist Richard Feynman originated a new way of thinking

about charge: a charge of a certain type is equivalent to a charge

of the opposite type that happens to be moving backward in time!

An electron moving backward in time is an antielectron — a par-

ticle that has the same mass as an electron, but whose charge is

opposite. Likewise we have antiprotons, and antimatter made from

antiprotons and antielectrons. Antielectrons occur naturally every-

where around you due to natural radioactive decay and radiation

from outer space. A small number of antihydrogen atoms has even

been created in particle accelerators!

Show that, for each rule for magnetic interactions shown in m, the

rule is still valid if you replace one of the charges with an opposite

charge moving in the opposite direction (i.e., backward in time).

4

Refer to figure r on page 121. Electrons have the type of charge

I’ve been representing with light-colored spheres.

(a) As the electrons in the beam pass over the top of the circle,

what is the direction of the force on them? Use what you know

about circular motion.

(b) From this information, use figure q on page 121 to determine

the direction of the magnetic field (left, right, up, down, into the

page, or out of the page).

5

You can’t use a light wave to see things that are smaller than

the wavelength of the light.

(a) Referring to figure z on page 127, what color of light do you

think would be the best to use for microscopy?

(b) The size of an atom is about 10−10 meters. Can visible light be

128

Chapter 6

Fields

used to make images of individual atoms?

6

You know how a microwave gets some parts of your food hot,

but leaves other parts cold? Suppose someone is trying to convince

you of the following explanation for this fact: The microwaves inside

the oven form a stationary wave pattern, like the vibrations of a

clothesline or a guitar string. The food is heated unevenly because

the wave crests are a certain distance apart, and the parts of the food

that get heated the most are the ones where there’s a crest in the wave

pattern. Use the wavelength scale in figure z on page 127 as a way of

checking numerically whether this is a reasonable explanation.

7

This book begins and ends with the topic of light. Give an

example of how the correspondence principle applies here, referring

to a concrete observation from a lab.

Stationary

wave

patterns

on

a clothesline (problem 6).

Problems

129

130

Chapter 6

Fields

Chapter 7

The Ray Model of Light

7.1 Light rays

Ads for one Macintosh computer bragged that it could do an arith-

metic calculation in less time than it took for the light to get from the

screen to your eye. We find this impressive because of the contrast

between the speed of light and the speeds at which we interact with

physical objects in our environment. Perhaps it shouldn’t surprise

us, then, that Newton succeeded so well in explaining the motion of

objects, but was far less successful with the study of light.

The climax of our study of electricity and magnetism was discov-

ery that light is an electromagnetic wave. Knowing this, however, is

not the same as knowing everything about eyes and telescopes. In

fact, the full description of light as a wave can be rather cumber-

some. In this chapter, we’ll instead make use of a simpler model of

light, the ray model, which does a fine job in most practical situa-

tions. Not only that, but we’ll even backtrack a little and start with

a discussion of basic ideas about light and vision that predated the

discovery of electromagnetic waves.

131

The nature of light

The cause and effect relationship in vision

Despite its title, this chapter is far from your first look at light.

That familiarity might seem like an advantage, but most people have

never thought carefully about light and vision. Even smart people

who have thought hard about vision have come up with incorrect

ideas. The ancient Greeks, Arabs and Chinese had theories of light

and vision, all of which were mostly wrong, and all of which were

accepted for thousands of years.

One thing the ancients did get right is that there is a distinction

between objects that emit light and objects that don’t. When you

see a leaf in the forest, it’s because three different objects are doing

their jobs: the leaf, the eye, and the sun. But luminous objects

like the sun, a flame, or the filament of a light bulb can be seen by

the eye without the presence of a third object. Emission of light

is often, but not always, associated with heat. In modern times,

we are familiar with a variety of objects that glow without being

heated, including fluorescent lights and glow-in-the-dark toys.

How do we see luminous objects? The Greek philosophers Pythago-

ras (b.

ca.

560 BC) and Empedocles of Acragas (b.

ca.

492

BC), who unfortunately were very influential, claimed that when

you looked at a candle flame, the flame and your eye were both

sending out some kind of mysterious stuff, and when your eye’s stuff

collided with the candle’s stuff, the candle would become evident to

your sense of sight.

Bizarre as the Greek “collision of stuff theory” might seem, it

had a couple of good features. It explained why both the candle

and your eye had to be present for your sense of sight to function.

The theory could also easily be expanded to explain how we see

nonluminous objects. If a leaf, for instance, happened to be present

at the site of the collision between your eye’s stuff and the candle’s

stuff, then the leaf would be stimulated to express its green nature,

allowing you to perceive it as green.

Modern people might feel uneasy about this theory, since it sug-

gests that greenness exists only for our seeing convenience, implying

a human precedence over natural phenomena. Nowadays, people

would expect the cause and effect relationship in vision to be the

other way around, with the leaf doing something to our eye rather

than our eye doing something to the leaf. But how can you tell?

The most common way of distinguishing cause from effect is to de-

termine which happened first, but the process of seeing seems to

occur too quickly to determine the order in which things happened.

Certainly there is no obvious time lag between the moment when

you move your head and the moment when your reflection in the

mirror moves.

132

Chapter 7

The Ray Model of Light

Today, photography provides the simplest experimental evidence

that nothing has to be emitted from your eye and hit the leaf in order

to make it “greenify.” A camera can take a picture of a leaf even

if there are no eyes anywhere nearby. Since the leaf appears green

regardless of whether it is being sensed by a camera, your eye, or

an insect’s eye, it seems to make more sense to say that the leaf’s

greenness is the cause, and something happening in the camera or

eye is the effect.

Light is a thing, and it travels from one point to another.

Another issue that few people have considered is whether a can-

dle’s flame simply affects your eye directly, or whether it sends out

light which then gets into your eye. Again, the rapidity of the effect

makes it difficult to tell what’s happening. If someone throws a rock

at you, you can see the rock on its way to your body, and you can

tell that the person affected you by sending a material substance

your way, rather than just harming you directly with an arm mo-

tion, which would be known as “action at a distance.” It is not easy

to do a similar observation to see whether there is some “stuff” that

travels from the candle to your eye, or whether it is a case of action

a / Light from a candle is bumped

at a distance.

off course by a piece of glass.

Inserting the glass causes the

Our description of the physics of material objects included both

apparent location of the candle

action at a distance (e.g., the earth’s gravitational force on a falling

to shift.

The same effect can

object) and contact forces such as friction.

be produced by taking off your

eyeglasses and looking at which

One piece of evidence that the candle sends out stuff that travels

you see near the edge of the

to your eye is that as in figure a, intervening transparent substances

lens, but a flat piece of glass

can make the candle appear to be in the wrong location, suggesting

works just as well as a lens for

that light is a thing that can be bumped off course. Many peo-

this purpose.

ple would dismiss this kind of observation as an optical illusion,

however. (Some optical illusions are purely neurological or psycho-

logical effects, although some others, including this one, turn out to

be caused by the behavior of light itself.)

A more convincing way to decide in which category light belongs

is to find out if it takes time to get from the candle to your eye; in

Newton’s picture of the universe, action at a distance was supposed

to be instantaneous. The fact that we speak casually today of “the

speed of light” implies that at some point in history, somebody

succeeded in showing that light did not travel infinitely fast. Galileo

tried, and failed, to detect a finite speed for light, by arranging with

a person in a distant tower to signal back and forth with lanterns.

Galileo uncovered his lantern, and when the other person saw the

light, he uncovered his lantern. Galileo was unable to measure any

time lag that was significant compared to the limitations of human

reflexes.

The first person to prove that light’s speed was finite, and to

determine it numerically, was Ole Roemer, in a series of measure-

Section 7.1

Light rays

133

ments around the year 1675. Roemer observed Io, one of Jupiter’s

moons, over a period of several years. Since Io presumably took the

same amount of time to complete each orbit of Jupiter, it could be

thought of as a very distant, very accurate clock. A practical and ac-

curate pendulum clock had recently been invented, so Roemer could

check whether the ratio of the two clocks’ cycles, about 42.5 hours

to 1 orbit, stayed exactly constant or changed a little. If the process

of seeing the distant moon was instantaneous, there would be no

reason for the two to get out of step. Even if the speed of light was

b / An

image

of

Jupiter

and

finite, you might expect that the result would be only to offset one

its moon Io (left) from the Cassini

cycle relative to the other. The earth does not, however, stay at a

probe.

constant distance from Jupiter and its moons. Since the distance is

changing gradually due to the two planets’ orbital motions, a finite

speed of light would make the “Io clock” appear to run faster as the

planets drew near each other, and more slowly as their separation

increased. Roemer did find a variation in the apparent speed of Io’s

orbits, which caused Io’s eclipses by Jupiter (the moments when Io

passed in front of or behind Jupiter) to occur about 7 minutes early

when the earth was closest to Jupiter, and 7 minutes late when it

was farthest. Based on these measurements, Roemer estimated the

speed of light to be approximately 2 × 108 m/s, which is in the right

ballpark compared to modern measurements of 3×108 m/s. (I’m not

sure whether the fairly large experimental error was mainly due to

c / The

earth

is

moving

to-

imprecise knowledge of the radius of the earth’s orbit or limitations

ward Jupiter and Io.

Since the

in the reliability of pendulum clocks.)

distance is shrinking, it is taking

less and less time for the light to

Light can travel through a vacuum.

get to us from Io, and Io appears

to circle Jupiter more quickly than

Many people are confused by the relationship between sound

normal.

Six months later, the

and light. Although we use different organs to sense them, there are

earth will be on the opposite side

some similarities. For instance, both light and sound are typically

of the sun, and receding from

emitted in all directions by their sources. Musicians even use visual

Jupiter and Io, so Io will appear

metaphors like “tone color,” or “a bright timbre” to describe sound.

to revolve around Jupiter more

One way to see that they are clearly different phenomena is to note

slowly.

their very different velocities. Sure, both are pretty fast compared

to a flying arrow or a galloping horse, but as we’ve seen, the speed of

light is so great as to appear instantaneous in most situations. The

speed of sound, however, can easily be observed just by watching a

group of schoolchildren a hundred feet away as they clap their hands

to a song. There is an obvious delay between when you see their

palms come together and when you hear the clap.

The fundamental distinction between sound and light is that

sound is an oscillation in air pressure, so it requires air (or some

other medium such as water) in which to travel. Today, we know

that outer space is a vacuum, so the fact that we get light from the

sun, moon and stars clearly shows that air is not necessary for the

propagation of light.

134

Chapter 7

The Ray Model of Light

Interaction of light with matter

Absorption of light

The reason why the sun feels warm on your skin is that the

sunlight is being absorbed, and the light energy is being transformed

into heat energy. The same happens with artificial light, so the net

result of leaving a light turned on is to heat the room. It doesn’t

matter whether the source of the light is hot, like the sun, a flame,

or an incandescent light bulb, or cool, like a fluorescent bulb. (If

your house has electric heat, then there is absolutely no point in

fastidiously turning off lights in the winter; the lights will help to

heat the house at the same dollar rate as the electric heater.)

This process of heating by absorption is entirely different from

heating by thermal conduction, as when an electric stove heats

spaghetti sauce through a pan. Heat can only be conducted through

matter, but there is vacuum between us and the sun, or between us

and the filament of an incandescent bulb. Also, heat conduction can

only transfer heat energy from a hotter object to a colder one, but a

cool fluorescent bulb is perfectly capable of heating something that

had already started out being warmer than the bulb itself.

How we see nonluminous objects

Not all the light energy that hits an object is transformed into

heat. Some is reflected, and this leads us to the question of how

we see nonluminous objects. If you ask the average person how we

see a light bulb, the most likely answer is “The light bulb makes

light, which hits our eyes.” But if you ask how we see a book, they

are likely to say “The bulb lights up the room, and that lets me

see the book.” All mention of light actually entering our eyes has

mysteriously disappeared.

Most people would disagree if you told them that light was re-

flected from the book to the eye, because they think of reflection as

something that mirrors do, not something that a book does. They

associate reflection with the formation of a reflected image, which

does not seem to appear in a piece of paper.

d / Two

self-portraits

of

the

author, one taken in a mirror and

Imagine that you are looking at your reflection in a nice smooth

one with a piece of aluminum foil.

piece of aluminum foil, fresh off the roll. You perceive a face, not a

piece of metal. Perhaps you also see the bright reflection of a lamp

over your shoulder behind you. Now imagine that the foil is just

a little bit less smooth. The different parts of the image are now

a little bit out of alignment with each other. Your brain can still

recognize a face and a lamp, but it’s a little scrambled, like a Picasso

painting. Now suppose you use a piece of aluminum foil that has

been crumpled up and then flattened out again. The parts of the

image are so scrambled that you cannot recognize an image. Instead,

e / Specular

and

diffuse

re-

your brain tells you you’re looking at a rough, silvery surface.

flection.

Section 7.1

Light rays

135

Mirror-like reflection at a specific angle is known as specular

reflection, and random reflection in many directions is called diffuse

reflection.

Diffuse reflection is how we see nonluminous objects.

Specular reflection only allows us to see images of objects other

than the one doing the reflecting. In top part of figure d, imagine

that the rays of light are coming from the sun. If you are looking

down at the reflecting surface, there is no way for your eye-brain

system to tell that the rays are not really coming from a sun down

below you.

Figure f shows another example of how we can’t avoid the con-

clusion that light bounces off of things other than mirrors. The

lamp is one I have in my house. It has a bright bulb, housed in a

completely opaque bowl-shaped metal shade. The only way light

can get out of the lamp is by going up out of the top of the bowl.

The fact that I can read a book in the position shown in the figure

means that light must be bouncing off of the ceiling, then bouncing

off of the book, then finally getting to my eye.

This is where the shortcomings of the Greek theory of vision

become glaringly obvious. In the Greek theory, the light from the

bulb and my mysterious “eye rays” are both supposed to go to the

book, where they collide, allowing me to see the book. But we now

have a total of four objects: lamp, eye, book, and ceiling. Where

does the ceiling come in? Does it also send out its own mysterious

“ceiling rays,” contributing to a three-way collision at the book?

That would just be too bizarre to believe!

The differences among white, black, and the various shades of

f / Light

bounces

off

of

the

gray in between is a matter of what percentage of the light they

ceiling, then off of the book.

absorb and what percentage they reflect. That’s why light-colored

clothing is more comfortable in the summer, and light-colored up-

holstery in a car stays cooler that dark upholstery.

The ray model of light

Models of light

Note how I’ve been casually diagramming the motion of light

with pictures showing light rays as lines on the page. Figure g shows

some more examples. More formally, this is known as the ray model

of light. The ray model of light seems natural once we convince

ourselves that light travels through space, and observe phenomena

like sunbeams coming through holes in clouds. If you’ve read chap-

ter 6, you’ve already been introduced to the concept of light as an

electromagnetic wave, and you know that the ray model is not the

ultimate truth about light, but the ray model is simpler, and in any

case science always deals with models of reality, not the ultimate

nature of reality. Figure h summarizes three models of light.

136

Chapter 7

The Ray Model of Light

g / Examples of ray diagrams.

h / Three models of light.

The ray model is a generic one. By using it we can discuss the

path taken by the light, without committing ourselves to any specific

description of what it is that is moving along that path. We will

use the nice simple ray model for rest of this chapter, and with it

we can analyze a great many devices and phenomena.

Note that the statements about the applicability of the various

models are only rough guides. For instance, wave interference effects

are often detectable, if small, when light passes around an obstacle

that is quite a bit bigger than a wavelength. Also, the criterion for

when we need the particle model really has more to do with energy

scales than distance scales, although the two turn out to be related.

The alert reader may have noticed that the wave model is re-

quired at scales smaller than a wavelength of light (on the order of a

micrometer for visible light), and the particle model is demanded on

the atomic scale or lower (a typical atom being a nanometer or so in

size). This implies that at the smallest scales we need both the wave

model and the particle model. They appear incompatible, so how

can we simultaneously use both? The answer is that they are not

as incompatible as they seem. Light is both a wave and a particle,

Section 7.1

Light rays

137

and a full understanding of this apparently nonsensical statement

would lead us to a discussion of the quantum physics revolution of

the twentieth century.

Geometry of specular reflection

Specular reflection obeys two simple geometrical rules:

• The angle of the reflected ray is the same as that of the incident

ray.

• The reflected ray lies in the plane containing the incident ray

and the normal (perpendicular) line. This plane is known as

the plane of incidence.

The two angles can be defined either with respect to the normal,

i

like angles B and C in the figure, or with respect to the reflecting

/ The

geometry

of

specular

reflection.

surface, like angles A and D. There is a convention of several hundred

years’ standing that one measures the angles with respect to the

normal, but the rule about equal angles can logically be stated either

as B=C or as A=D.

self-check A

Each of these diagrams is supposed to show two different rays being

reflected from the same point on the same mirror. Which are correct,

and which are incorrect?

. Answer, p. 180

Ray diagrams

Figure j shows some guidelines for using ray diagrams effectively.

The light rays bend when then pass out through the surface of the

water (a phenomenon that we’ll discuss in more detail later). The

rays appear to have come from a point above the goldfish’s actual

location, an effect that is familiar to people who have tried spear-

fishing.

• A stream of light is not really confined to a finite number of

narrow lines. We just draw it that way. In j/1, it has been

necessary to choose a finite number of rays to draw (five),

rather than the theoretically infinite number of rays that will

diverge from that point.

• There is a tendency to conceptualize rays incorrectly as ob-

jects. In his Optics, Newton goes out of his way to caution

the reader against this, saying that some people “consider ...

the refraction of ... rays to be the bending or breaking of them

in their passing out of one medium into another.” But a ray

138

Chapter 7

The Ray Model of Light

is a record of the path traveled by light, not a physical thing

that can be bent or broken.

• In theory, rays may continue infinitely far into the past and

future, but we need to draw lines of finite length. In j/1, a

judicious choice has been made as to where to begin and end

the rays. There is no point in continuing the rays any farther

than shown, because nothing new and exciting is going to

happen to them. There is also no good reason to start them

earlier, before being reflected by the fish, because the direction

of the diffusely reflected rays is random anyway, and unrelated

to the direction of the original, incoming ray.

• When representing diffuse reflection in a ray diagram, many

students have a mental block against drawing many rays fan-

ning out from the same point. Often, as in example j/2, the

problem is the misconception that light can only be reflected

in one direction from one point.

• Another difficulty associated with diffuse reflection, example

j/3, is the tendency to think that in addition to drawing many

rays coming out of one point, we should also be drawing many

rays coming from many points. In j/1, drawing many rays

coming out of one point gives useful information, telling us,

for instance, that the fish can be seen from any angle. Drawing

many sets of rays, as in j/3, does not give us any more useful

information, and just clutters up the picture in this example.

The only reason to draw sets of rays fanning out from more

than one point would be if different things were happening to

the different sets.

j / 1. Correct. 2. Incorrect: im-

plies that diffuse reflection only

gives one ray from each reflecting

point. 3. Correct, but unneces-

sarily complicated

Section 7.1

Light rays

139

Discussion question

A

If you observe thunder and lightning, you can tell how far away the

storm is. Do you need to know the speed of sound, of light, or of both?

B

When phenomena like X-rays and cosmic rays were first discovered,

suggest a way one could have tested whether they were forms of light.

C

Why did Roemer only need to know the radius of the earth’s orbit,

not Jupiter’s, in order to find the speed of light?

D

The curtains in a room are drawn, but a small gap lets light through,

illuminating a spot on the floor. It may or may not also be possible to see

the beam of sunshine crossing the room, depending on the conditions.

What’s going on?

E

Laser beams are made of light. In science fiction movies, laser

beams are often shown as bright lines shooting out of a laser gun on a

spaceship. Why is this scientifically incorrect?

F

Suppose an intelligent tool-using fish is spear-hunting for humans.

Draw a ray diagram to show how the fish has to correct its aim. Note

that although the rays are now passing from the air to the water, the same

rules apply: the rays are closer to being perpendicular to the surface when

they are in the water, and rays that hit the air-water interface at a shallow

angle are bent the most.

7.2 Applications

The inverse-square law

Energy is conserved, so a ray of light should theoretically be

able to cross an infinite distance without losing any of its intensity,

provided that it’s traveling through empty space, so that there’s no

matter that it can give its energy away to. In that case, why does a

distant candle appear dim? Likewise, our sun is just a star like any

other star, but it appears much brighter because it’s so much closer

to us. Why are the other stars so dim if not because their light gets

“tired,” or “wears out?” It’s not that the light rays are stopping,

it’s that they’re getting spread out more thinly. The light comes

out of the source in all directions, and if you’re very far away, only

a tiny percentage of the light will go into your eye. (If all the light

from a star went into your eye, you’d be in trouble.)

k / The light is four times dimmer

at twice the distance.

140

Chapter 7

The Ray Model of Light

Figure k shows what happens if you double your distance from

the source. The light from the flame spreads out in all directions.

We pick four representative rays from among those that happen

to pass through the nearer square. Of these four, only one passes

through the square of equal area at twice the distance. If the two

equal-area squares were people’s eyes, then only one fourth of the

light would go into the more distant person’s eye.

Another way of thinking about it is that the light that passed

through the first square spreads out and makes a bigger square; at

double the distance, the square is twice as wide and twice as tall, so

its area is 2 × 2 = 4 times greater. The same light has been spread

out over four times the area.

In general, the rule works like this:

1

distance × 2 ⇒ brightness × 4

1

distance × 3 ⇒ brightness × 9

1

distance × 4 ⇒ brightness × 16

To get the 4, we multiplied 2 by itself, 9 came from multiplying 3 by

itself, and so on. Multiplying a number by itself is called squaring

it, and dividing one by a number is called inverting it, so a rela-

tionship like this is known as an inverse square law. Inverse square

laws are very common in physics: they occur whenever something is

spreading out in all directions from a point. Physicists already knew

about this kind of inverse square law, for light, before Newton found

out that the force of gravity varied as an inverse square, so his law

of gravity made sense to them intuitively, and they were ready to

accept it. However, Newton’s law of gravity doesn’t describe grav-

ity as a substance that physically travels outward through space, so

it’s only a rough analogy. (One modern hypothesis about gravity

is that the messages of gravitational attraction between two objects

are actually carried by little particles, called gravitons, but nobody

has ever detected a graviton directly.)

self-check B

Alice is one meter from the candle, while Bob is at a distance of five

meters. How many times dimmer is the light at Bob’s location?

.

Answer, p. 180

An example with sound

example 1

. Four castaways are adrift in an open boat, and are yelling to try

to attract the attention of passing ships. If all four of them yell at

once, how much is their range increased compared to the range

they would have if they took turns yelling one at a time?

. This is an example involving sound. Although sound isn’t the

same as light, it does spread out in all directions from a source,

so it obeys the inverse-square law. In the previous examples, we

Section 7.2

Applications

141

knew the distance and wanted to find the intensity (brightness).

Here, we know about the intensity (loudness), and we want to

find out about the distance. Rather than taking a number and

multiplying it by itself to find the answer, we need to reverse the

process, and find the number that, when multiplied by itself, gives

four. In other words, we’re computing the square root of four,

which is two. They will double their range, not quadruple it.

Astronomical distance scales

example 2

The nearest star, Alpha Centauri,1 is about 10,000,000,000,000,000

times dimmer than our sun when viewed from our planet. If we as-

sume that Alpha Centauri’s true brightness is roughly the same

as that of our own sun, then we can find the distance to Alpha

Centauri by taking the square root of this number. Alpha Cen-

tauri’s distance from us is equal to about 100,000,000 times our

distance from the sun.

Pupils and camera diaphragms

example 3

In bright sunlight, your pupils contract to admit less light. At night

they dilate, becoming bigger “light buckets.” Your perception of

brightness depends not only on the true brightness of the source

and your distance from it, but also on how much area your pupils

present to the light. Cameras have a similar mechanism, which

is easy to see if you detach the lens and its housing from the

body of the camera, as shown in the figure. Here, the diameter

of the largest aperture is about ten times greater than that of the

smallest aperture. Making a circle ten times greater in radius

increases its area by a factor of 100, so the light-gathering power

of the camera becomes 100 times greater. (Many people expect

that the area would only be ten times greater, but if you start

drawing copies of the small circle inside the large circle, you’ll

see that ten are not nearly enough to fill in the entire area of the

larger circle. Both the width and the height of the bigger circle are

ten times greater, so its area is 100 times greater.)

l / The

same

lens

is

shown

with its diaphragm set to three

Parallax

different apertures.

Example 2 on page 142 showed how we can use brightness to de-

termine distance, but your eye-brain system has a different method.

Right now, you can tell how far away this page is from your eyes.

This sense of depth perception comes from the fact that your two

eyes show you the same scene from two different perspectives. If

you wink one eye and then the other, the page will appear to shift

back and forth a little.

If you were looking at a fly on the bridge of your nose, there

would be an angle of nearly 180◦ between the ray that went into

1Sticklers will note that the nearest star is really our own sun, and the second

nearest is the burned-out cinder known as Proxima Centauri, which is Alpha

Centauri’s close companion.

142

Chapter 7

The Ray Model of Light

m / At double the distance, the parallax angle is approximately halved.

your left eye and the one that went into your right. Your brain

would know that this large angle implied a very small distance.

This is called the parallax angle. Objects at greater distances have

smaller parallax angles, and when the angles are small, it’s a good

approximation to say that the angle is inversely proportional to the

distance. In figure m, the parallax angle is almost exactly cut in

half when the person moves twice as far away.

Parallax can be observed in other ways than with a pair of eye-

balls. As a child, you noticed that when you walked around on a

moonlit evening, the moon seemed to follow you. The moon wasn’t

really following you, and this isn’t even a special property of the

moon. It’s just that as you walk, you expect to observe a paral-

lax angle between the same scene viewed from different positions

of your whole head. Very distant objects, including those on the

Earth’s surface, have parallax angles too small to notice by walking

back and forth. In general, rays coming from a very distant object

are nearly parallel.

If your baseline is long enough, however, the small parallaxes

of even very distant objects may be detectable. In the nineteenth

century, nobody knew how tall the Himalayas were, or exactly where

their peaks were on a map, and the Andes were generally believed

to be the tallest mountains in the world. The Himalayas had never

been climbed, and could only be viewed from a distance. From down

on the plains of India, there was no way to tell whether they were

very tall mountains very far away, or relatively low ones that were

much closer.

British surveyor George Everest finally established

their true distance, and astounding height, by observing the same

peaks through a telescope from different locations far apart.

An even more spectacular feat of measurement was carried out

by Hipparchus over twenty-one centuries ago. By measuring the

parallax of the moon as observed from Alexandria and the Helle-

spont, he determined its distance to be about 90 times the radius

of the earth.2

2The reason this was a hard measurement was that accurate clocks hadn’t

been invented, so there was no easy way to synchronize the two observations,

and the desired effect would be masked by the apparent motion of the moon

across the sky as it rose and set. Hipparchus’s trick was to do the measurement

Section 7.2

Applications

143

The earth circles the sun, n, and we can therefore determine

the distances to a few hundred of the nearest stars by making ob-

servations six months apart, so that the baseline for the parallax

measurement is the diameter of the earth’s orbit. For these stars,

the distances derived from parallax can be checked against the ones

found by the method of example 2 on page 142. They do check out,

which verifies the assumption that the stars are objects analogous

to our sun.

n / The nearer star has a larger

parallax angle. By measuring the

parallax angles, we can deter-

mine the distances to both stars.

(The scale on this drawing is

not realistic.

If the earth’s orbit

was really this size, the nearest

stars would be several kilometers

away.)

Reversibility of light rays

The fact that specular reflection displays equal angles of inci-

dence and reflection means that there is a symmetry: if the ray had

come in from the right instead of the left in the figure above, the an-

gles would have looked exactly the same. This is not just a pointless

detail about specular reflection. It’s a manifestation of a very deep

and important fact about nature, which is that the laws of physics

do not distinguish between past and future. Cannonballs and plan-

ets have trajectories that are equally natural in reverse, and so do

light rays. This type of symmetry is called time-reversal symmetry.

Typically, time-reversal symmetry is a characteristic of any pro-

cess that does not involve heat. For instance, the planets do not

experience any friction as they travel through empty space, so there

is no frictional heating. We should thus expect the time-reversed

versions of their orbits to obey the laws of physics, which they do.

In contrast, a book sliding across a table does generate heat from

friction as it slows down, and it is therefore not surprising that this

type of motion does not appear to obey time-reversal symmetry. A

book lying still on a flat table is never observed to spontaneously

start sliding, sucking up heat energy and transforming it into kinetic

energy.

Similarly, the only situation we’ve observed so far where light

does not obey time-reversal symmetry is absorption, which involves

heat. Your skin absorbs visible light from the sun and heats up,

but we never observe people’s skin to glow, converting heat energy

into visible light.

People’s skin does glow in infrared light, but

that doesn’t mean the situation is symmetric. Even if you absorb

during a solar eclipse, so that people at both locations would know they were in

sync.

144

Chapter 7

The Ray Model of Light

infrared, you don’t emit visible light, because your skin isn’t hot

enough to glow in the visible spectrum.

Ray tracing on a computer

example 4

A number of techniques can be used for creating artificial visual

scenes in computer graphics. Figure o shows such a scene,

which was created by the brute-force technique of simply con-

structing a very detailed ray diagram on a computer. This tech-

nique requires a great deal of computation, and is therefore too

slow to be used for video games and computer-animated movies.

One trick for speeding up the computation is to exploit the re-

versibility of light rays. If one was to trace every ray emitted by

every illuminated surface, only a tiny fraction of those would actu-

ally end up passing into the virtual “camera,” and therefore almost

all of the computational effort would be wasted. One can instead

start a ray at the camera, trace it backward in time, and see where

it would have come from. With this technique, there is no wasted

effort.

p / Discussion

question

B.

o / This photorealistic image of a nonexistent countertop was pro-

duced completely on a computer, by computing a complicated ray

diagram.

Discussion questions

A

If a light ray has a velocity vector with components cx and cy , what

will happen when it is reflected from a surface that lies along the y axis?

q / Discussion

question

C.

Make sure your answer does not imply a change in the ray’s speed.

B

Generalizing your reasoning from discussion question A, what will

Section 7.2

Applications

145

happen to the velocity components of a light ray that hits a corner, as

shown in the figure, and undergoes two reflections?

C

Three pieces of sheet metal arranged perpendicularly as shown in

the figure form what is known as a radar corner. Let’s assume that the

radar corner is large compared to the wavelength of the radar waves, so

that the ray model makes sense. If the radar corner is bathed in radar

rays, at least some of them will undergo three reflections. Making a fur-

ther generalization of your reasoning from the two preceding discussion

questions, what will happen to the three velocity components of such a

ray? What would the radar corner be useful for?

r / The

solid

lines

are

physi-

cally possible paths for light rays

traveling from A to B and from

7.3 ? The principle of least time for reflection

A to C. They obey the principle

of least time. The dashed lines

There is another way of stating the rules of reflection that is very

do not obey the principle of

simple and beautiful, and turns out to have deep consequences and

least time, and are not physically

apply much more broadly, not just to reflection. It is called the

possible.

principle of least time, or Fermat’s principle.

Let’s start with the motion of light that is not interacting with

matter at all. In a vacuum, a light ray moves in a straight line. This

can be rephrased as follows: of all the conceivable paths light could

follow from P to Q, the only one that is physically possible is the

path that takes the least time.

What about reflection? If light is going to go from one point to

another, being reflected on the way, the quickest path is indeed the

one with equal angles of incidence and reflection. If the starting and

ending points are equally far from the reflecting surface, r, it’s not

hard to convince yourself that this is true, just based on symmetry.

There is also a tricky and simple proof, shown in figure s, for the

more general case where the points are at different distances from

s / Paths

AQB

and

APB

are

the surface.

two conceivable paths that a ray

could follow to get from A to B

Not only does the principle of least time work for light in a

with one reflection, but only AQB

vacuum and light undergoing reflection, we will also see in a later

is physically possible.

We wish

chapter that it works for the bending of light when it passes from

to prove that the path AQB, with

one medium into another.

equal angles of incidence and

reflection, is shorter than any

Although it is beautiful that the entire ray model of light can

other path, such as APB. The

be reduced to one simple rule, the principle of least time, it may

trick is to construct a third point,

seem a little spooky to speak as if the ray of light is intelligent,

C, lying as far below the surface

and has carefully planned ahead to find the shortest route to its

as B lies above it.

Then path

destination. How does it know in advance where it’s going? What

AQC is a straight line whose

length is the same as AQB’s, and

if we moved the mirror while the light was en route, so conditions

path APC has the same length as

along its planned path were not what it “expected?” The answer is

path APB. Since AQC is straight,

that the principle of least time is really an approximate shortcut for

it must be shorter than any other

finding certain results of the wave model of light.

path such as APC that connects

A and C, and therefore AQB must

There are a couple of subtle points about the principle of least

be shorter than any path such as

time. First, the path does not have to be the quickest of all pos-

APB.

sible paths; it only needs to be quicker than any path that differs

146

Chapter 7

The Ray Model of Light

infinitesimally from it. In figure s, for instance, light could get from

A to B either by the reflected path AQB or simply by going straight

from A to B. Although AQB is not the shortest possible path, it

cannot be shortened by changing it infinitesimally, e.g., by moving

Q a little to the right or left. On the other hand, path APB is phys-

ically impossible, because it is possible to improve on it by moving

point P infinitesimally to the right.

It’s not quite right to call this the principle of least time. In fig-

ure t, for example, the four physically possible paths by which a ray

can return to the center consist of two shortest-time paths and two

longest-time paths. Strictly speaking, we should refer to the prin-

ciple of least or greatest time, but most physicists omit the niceties,

and assume that other physicists understand that both maxima and

minima are possible.

t / Light is emitted at the center

of an elliptical mirror. There are

four physically possible paths by

7.4 Images by reflection

which a ray can be reflected and

Infants are always fascinated by the antics of the Baby in the Mirror.

return to the center.

Now if you want to know something about mirror images that most

people don’t understand, try this. First bring this page closer wand

closer to your eyes, until you can no longer focus on it without

straining.

Then go in the bathroom and see how close you can

get your face to the surface of the mirror before you can no longer

easily focus on the image of your own eyes.

You will find that

the shortest comfortable eye-mirror distance is much less than the

shortest comfortable eye-paper distance. This demonstrates that

the image of your face in the mirror acts as if it had depth and

existed in the space behind the mirror. If the image was like a flat

picture in a book, then you wouldn’t be able to focus on it from

such a short distance.

In this chapter we will study the images formed by flat and

curved mirrors on a qualitative, conceptual basis. Although this

type of image is not as commonly encountered in everyday life as

images formed by lenses, images formed by reflection are simpler to

understand.

A virtual image

We can understand a mirror image using a ray diagram. Figure

u shows several light rays, 1, that originated by diffuse reflection at

the person’s nose. They bounce off the mirror, producing new rays,

2. To anyone whose eye is in the right position to get one of these

rays, they appear to have come from a behind the mirror, 3, where

they would have originated from a single point. This point is where

the tip of the image-person’s nose appears to be. A similar analysis

applies to every other point on the person’s face, so it looks as

u / An

image

formed

by

a

though there was an entire face behind the mirror. The customary

mirror.

way of describing the situation requires some explanation:

Section 7.4

Images by reflection

147

Customary description in physics: There is an image of the face

behind the mirror.

Translation: The pattern of rays coming from the mirror is exactly

the same as it would be if there was a face behind the mirror.

Nothing is really behind the mirror.

This is referred to as a virtual image, because the rays do not

actually cross at the point behind the mirror. They only appear to

have originated there.

self-check C

Imagine that the person in figure u moves his face down quite a bit — a

couple of feet in real life, or a few inches on this scale drawing. Draw a

new ray diagram. Will there still be an image? If so, where is it visible

from?

. Answer, p. 180

The geometry of specular reflection tells us that rays 1 and 2

are at equal angles to the normal (the imaginary perpendicular line

piercing the mirror at the point of reflection). This means that ray

2’s imaginary continuation, 3, forms the same angle with the mirror

as ray 1. Since each ray of type 3 forms the same angles with the

mirror as its partner of type 1, we see that the distance of the image

from the mirror is the same as the actual face from the mirror, and

lies directly across from it. The image therefore appears to be the

same size as the actual face.

Curved mirrors

An image in a flat mirror is a pretechnological example: even

animals can look at their reflections in a calm pond. We now pass

to our first nontrivial example of the manipulation of an image by

technology: an image in a curved mirror. Before we dive in, let’s

consider why this is an important example. If it was just a ques-

tion of memorizing a bunch of facts about curved mirrors, then you

would rightly rebel against an effort to spoil the beauty of your lib-

erally educated brain by force-feeding you technological trivia. The

reason this is an important example is not that curved mirrors are

so important in and of themselves, but that the results we derive for

curved bowl-shaped mirrors turn out to be true for a large class of

v / An

image

formed

by

a

other optical devices, including mirrors that bulge outward rather

curved mirror.

than inward, and lenses as well. A microscope or a telescope is sim-

ply a combination of lenses or mirrors or both. What you’re really

learning about here is the basic building block of all optical devices

from movie projectors to octopus eyes.

Because the mirror in figure v is curved, it bends the rays back

closer together than a flat mirror would: we describe it as converging.

Note that the term refers to what it does to the light rays, not to the

physical shape of the mirror’s surface . (The surface itself would be

148

Chapter 7

The Ray Model of Light

described as concave. The term is not all that hard to remember,

because the hollowed-out interior of the mirror is like a cave.) It

is surprising but true that all the rays like 3 really do converge on

a point, forming a good image. We will not prove this fact, but it

is true for any mirror whose curvature is gentle enough and that

is symmetric with respect to rotation about the perpendicular line

passing through its center (not asymmetric like a potato chip). The

old-fashioned method of making mirrors and lenses is by grinding

them in grit by hand, and this automatically tends to produce an

almost perfect spherical surface.

Bending a ray like 2 inward implies bending its imaginary contin-

uation 3 outward, in the same way that raising one end of a seesaw

causes the other end to go down. The image therefore forms deeper

behind the mirror. This doesn’t just show that there is extra dis-

tance between the image-nose and the mirror; it also implies that

the image itself is bigger from front to back. It has been magnified

in the front-to-back direction.

It is easy to prove that the same magnification also applies to the

image’s other dimensions. Consider a point like E in figure w. The

trick is that out of all the rays diffusely reflected by E, we pick the

one that happens to head for the mirror’s center, C. The equal-angle

property of specular reflection plus a little straightforward geometry

easily leads us to the conclusion that triangles ABC and CDE are

the same shape, with ABC being simply a scaled-up version of CDE.

The magnification of depth equals the ratio BC/CD, and the up-

down magnification is AB/DE. A repetition of the same proof shows

w / The

image

is

magnified

that the magnification in the third dimension (out of the page) is

by the same factor in depth and

also the same. This means that the image-head is simply a larger

in its other dimensions.

version of the real one, without any distortion. The scaling factor

is called the magnification, M . The image in the figure is magnified

by a factor M = 1.9.

Note that we did not explicitly specify whether the mirror was

a sphere, a paraboloid, or some other shape. However, we assumed

that a focused image would be formed, which would not necessarily

be true, for instance, for a mirror that was asymmetric or very deeply

curved.

A real image

If we start by placing an object very close to the mirror, x/1,

and then move it farther and farther away, the image at first behaves

as we would expect from our everyday experience with flat mirrors,

receding deeper and deeper behind the mirror. At a certain point,

however, a dramatic change occurs. When the object is more than

a certain distance from the mirror, x/2, the image appears upside-

down and in front of the mirror.

Here’s what’s happened. The mirror bends light rays inward, but

Section 7.4

Images by reflection

149

when the object is very close to it, as in x/1, the rays coming from a

given point on the object are too strongly diverging (spreading) for

the mirror to bring them back together. On reflection, the rays are

still diverging, just not as strongly diverging. But when the object

is sufficiently far away, x/2, the mirror is only intercepting the rays

that came out in a narrow cone, and it is able to bend these enough

so that they will reconverge.

Note that the rays shown in the figure, which both originated at

the same point on the object, reunite when they cross. The point

where they cross is the image of the point on the original object.

This type of image is called a real image, in contradistinction to the

virtual images we’ve studied before. The use of the word “real” is

perhaps unfortunate. It sounds as though we are saying the image

was an actual material object, which of course it is not.

x / 1.

A virtual image.

2.

A

real image.

As you’ll verify in

homework problem 9, the image

is upside-down

The distinction between a real image and a virtual image is an

important one, because a real image can be projected onto a screen

or photographic film. If a piece of paper is inserted in figure x/2

at the location of the image, the image will be visible on the paper

(provided the object is bright and the room is dark). Your eye uses

a lens to make a real image on the retina.

self-check D

Sketch another copy of the face in figure x/1, even farther from the mir-

150

Chapter 7

The Ray Model of Light

ror, and draw a ray diagram. What has happened to the location of the

image?

. Answer, p. 180

Images of images

If you are wearing glasses right now, then the light rays from the

page are being manipulated first by your glasses and then by the lens

of your eye. You might think that it would be extremely difficult

to analyze this, but in fact it is quite easy. In any series of optical

elements (mirrors or lenses or both), each element works on the rays

furnished by the previous element in exactly the same manner as if

the image formed by the previous element was an actual object.

Figure y shows an example involving only mirrors. The Newto-

nian telescope, invented by Isaac Newton, consists of a large curved

mirror, plus a second, flat mirror that brings the light out of the

tube. (In very large telescopes, there may be enough room to put

a camera or even a person inside the tube, in which case the sec-

ond mirror is not needed.) The tube of the telescope is not vital; it

is mainly a structural element, although it can also be helpful for

blocking out stray light. The lens has been removed from the front

of the camera body, and is not needed for this setup. Note that the

two sample rays have been drawn parallel, because an astronomical

telescope is used for viewing objects that are extremely far away.

These two “parallel” lines actually meet at a certain point, say a

crater on the moon, so they can’t actually be perfectly parallel, but

they are parallel for all practical purposes since we would have to

y / A

Newtonian

telescope

follow them upward for a quarter of a million miles to get to the

being used with a camera.

point where they intersect.

The large curved mirror by itself would form an image I, but the

small flat mirror creates an image of the image, I0. The relationship

between I and I0 is exactly the same as it would be if I was an actual

object rather than an image: I and I0 are at equal distances from

the plane of the mirror, and the line between them is perpendicular

to the plane of the mirror.

One surprising wrinkle is that whereas a flat mirror used by itself

forms a virtual image of an object that is real, here the mirror is

forming a real image of virtual image I. This shows how pointless it

would be to try to memorize lists of facts about what kinds of images

are formed by various optical elements under various circumstances.

You are better off simply drawing a ray diagram.

Although the main point here was to give an example of an image

z / A

Newtonian

telescope

of an image, figure z shows an interesting case where we need to make

being used for visual rather than

the distinction between magnification and angular magnification. If

photographic observing.

In real

you are looking at the moon through this telescope, then the images

life, an eyepiece lens is normally

I and I0 are much smaller than the actual moon. Otherwise, for

used for additional magnification,

example, image I would not fit inside the telescope! However, these

but this simpler setup will also

work.

images are very close to your eye compared to the actual moon. The

Section 7.4

Images by reflection

151

small size of the image has been more than compensated for by the

shorter distance. The important thing here is the amount of angle

within your field of view that the image covers, and it is this angle

that has been increased. The factor by which it is increased is called

the angular magnification, Ma.

aa / The angular size of the flower

depends on its distance from the

eye.

152

Chapter 7

The Ray Model of Light

Discussion questions

A

The figure shows an object that is off to one side of a mirror. Draw

a ray diagram. Is an image formed? If so, where is it, and from which

directions would it be visible?

B

Locate the images of you that will be formed if you stand between

two parallel mirrors.

Section 7.4

Images by reflection

153

C

Locate the images formed by two perpendicular mirrors, as in the

figure. What happens if the mirrors are not perfectly perpendicular?

154

Chapter 7

The Ray Model of Light

D

Locate the images formed by the periscope.

Section 7.4

Images by reflection

155

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

The natives of planet Wumpus play pool using light rays on

an eleven-sided table with mirrors for bumpers, shown in the figure

on the next page. Trace this shot accurately with a ruler to reveal

the hidden message. To get good enough accuracy, you’ll need to

photocopy the page (or download the book and print the page) and

construct each reflection using a protractor.

. Solution, p. ??

Problem 1.

2

The figure on the next page shows a curved (parabolic) mir-

ror, with three parallel light rays coming toward it.

One ray is

approaching along the mirror’s center line. (a) Continue the light

rays until they are about to undergo their second reflection. To get

good enough accuracy, you’ll need to photocopy the page (or down-

load the book and print the page) and draw in the normal at each

place where a ray is reflected. What do you notice? (b) Make up

an example of a practical use for this device. (c) How could you

use this mirror with a small lightbulb to produce a parallel beam of

light rays going off to the right?

. Solution, p. ??

3

The figure shows four points where rays cross. Of these, which

Problem 3.

are image points? Explain.

156

Chapter 7

The Ray Model of Light

Problem 2.

4

In this chapter we’ve only done examples of mirrors with

hollowed-out shapes (called concave mirrors). Now draw a ray dia-

gram for a curved mirror that has a bulging outward shape (called a

convex mirror). (a) How does the image’s distance from the mirror

compare with the actual object’s distance from the mirror? From

this comparison, determine whether the magnification is greater

than or less than one.

(b) Is the image real or virtual?

Could

this mirror ever make the other type of image?

5

Draw a ray diagram showing why a small light source (a

candle, say) produces sharper shadows than a large one (e.g., a long

fluorescent bulb).

6

A man is walking at 1.0 m/s directly towards a flat mirror.

√

At what speed is his separation from his image decreasing?

7

If a mirror on a wall is only big enough for you to see your-

self from your head down to your waist, can you see your entire

body by backing up? Test this experimentally and come up with an

explanation for your observations, including a ray diagram.

Note that when you do the experiment, it’s easy to confuse yourself

if the mirror is even a tiny bit off of vertical. One way to check

yourself is to artificially lower the top of the mirror by putting a

piece of tape or a post-it note where it blocks your view of the top

of your head. You can then check whether you are able to see more

of yourself both above and below by backing up.

8

As discussed in question 4, there are two types of curved

Problems

157

mirrors, concave and convex. Make a list of all the possible com-

binations of types of images (virtual or real) with types of mirrors

(concave and convex). (Not all of the four combinations are phys-

ically possible.) Now for each one, use ray diagrams to determine

whether increasing the distance of the object from the mirror leads

to an increase or a decrease in the distance of the image from the

mirror.

Draw BIG ray diagrams! Each diagram should use up about half a

page of paper.

Some tips: To draw a ray diagram, you need two rays. For one of

these, pick the ray that comes straight along the mirror’s axis, since

its reflection is easy to draw. After you draw the two rays and locate

the image for the original object position, pick a new object position

that results in the same type of image, and start a new ray diagram,

in a different color of pen, right on top of the first one. For the two

new rays, pick the ones that just happen to hit the mirror at the

same two places; this makes it much easier to get the result right

without depending on extreme accuracy in your ability to draw the

reflected rays.

9

In figure x/2 in on page 150, only the image of my forehead was

located by drawing rays. Either photocopy the figure or download

the book and print out the relevant page. On this copy of the figure,

make a new set of rays coming from my chin, and locate its image.

To make it easier to judge the angles accurately, draw rays from the

chin that happen to hit the mirror at the same points where the two

rays from the forehead were shown hitting it. By comparing the

locations of the chin’s image and the forehead’s image, verify that

the image is actually upside-down, as shown in the original figure.

10

If the user of an astronomical telescope moves her head

closer to or farther away from the image she is looking at, does

the magnification change? Does the angular magnification change?

Explain. (For simplicity, assume that no eyepiece is being used.)

. Solution, p. ??

11

Here’s a game my kids like to play. I sit next to a sunny

window, and the sun reflects from the glass on my watch, making a

disk of light on the wall or floor, which they pretend to chase as I

move it around. Is the spot a disk because that’s the shape of the

sun, or because it’s the shape of my watch? In other words, would

a square watch make a square spot, or do we just have a circular

image of the circular sun, which will be circular no matter what?

12

A Global Positioning System (GPS) receiver is a device that

lets you figure out where you are by measuring the time for radio

signals to travel between you and the satellite, which is related to

158

Chapter 7

The Ray Model of Light

the distance between you and the satellite. By finding the ranges to several different satellites in this way, it can pin down your location

in three dimensions to within a few meters. How accurate does the

measurement of the time delay have to be to determine your position

to this accuracy?

13

Estimate the frequency of an electromagnetic wave whose

wavelength is similar in size to an atom (about a nm). Referring

back to figure z on p. 127, in what part of the electromagnetic

spectrum would such a wave lie (infrared, gamma-rays, . . . )?

14

The Stealth Bomber is designed with flat, smooth surfaces.

Why would this make it difficult to detect using radar?

. Solution, p. ??

Problems

159

160

Chapter 7

The Ray Model of Light

“The Great Wave Off Kanagawa,” by Katsushika Hokusai (1760-1849).

Chapter 8

Waves

If you’ve read chapter 6, you’ve been introduced to the idea that

the universe isn’t really mechanical in nature. It’s made of fields

of force. When a radio antenna makes a disturbance in the electric

and magnetic fields, those disturbances travel outward like ripples

on a pond. In other words, waves are fundamental to the way the

universe works.

8.1 Vibrations

Your radio dial is calibrated in units of frequency, the simplest ex-

ample of this concept is provided not by a wave but by a vibrating

physical object such as a mass on the end of a spring, 8. With no

forces on it, the spring assumes its equilibrium length, 8/1. It can

be stretched, 2, or compressed, 3. We attach the spring to a wall

on the left and to a mass on the right. If we now hit the mass with

a hammer, 4, it oscillates as shown in the series of snapshots, 4-13.

If we assume that the mass slides back and forth without friction

a / A

spring

has

an

equilib-

and that the motion is one-dimensional, then conservation of energy

rium length,

1,

and can be

proves that the motion must be repetitive. When the block comes

stretched, 2, or compressed, 3. A

back to its initial position again, 7, its potential energy is the same

mass attached to the spring can

again, so it must have the same kinetic energy again. The motion

be set into motion initially, 4, and

will then vibrate, 4-13.

is in the opposite direction, however. Finally, at 10, it returns to its

161

initial position with the same kinetic energy and the same direction

of motion. The motion has gone through one complete cycle, and

will now repeat forever in the absence of friction.

The usual physics terminology for motion that repeats itself over

and over is periodic motion, and the time required for one repetition

is called the period, T . One complete repetition of the motion is

called a cycle.

We are used to referring to short-period sound vibrations as

“high” in pitch, and it sounds odd to have to say that high pitches

have low periods. It is therefore more common to discuss the rapid-

ity of a vibration in terms of the number of vibrations per second,

a quantity called the frequency, f . Since the period is the number

of seconds per cycle and the frequency is the number of cycles per

second, they are reciprocals of each other,

f = 1/T .

Units of inverse second, s−1, are awkward in speech, so an abbre-

viation has been created. One Hertz, named in honor of a pioneer

of radio technology, is one cycle per second. In abbreviated form,

1 Hz = 1 s−1. This is the familiar unit used for the frequencies on

the radio dial.

Frequency of a radio station

example 1

. KMHD’s frequency is 89.1 MHz. What does this mean, and

what period does this correspond to?

. The metric prefix M- is mega-, i.e., millions. The radio waves

emitted by KMHD’s transmitting antenna vibrate 89.1 million times

per second. This corresponds to a period of

T = 1/f = 1.12 × 10−8 s.

This example shows a second reason why we normally speak in

terms of frequency rather than period: it would be painful to have

to refer to such small time intervals routinely. I could abbreviate

by telling people that KMHD’s period was 11.2 nanoseconds, but

most people are more familiar with the big metric prefixes than

with the small ones.

Units of frequency are also commonly used to specify the speeds

of computers. The idea is that all the little circuits on a computer

b / 1.

The amplitude of the

vibrations of the mass on a spring

chip are synchronized by the very fast ticks of an electronic clock, so

could be defined in two different

that the circuits can all cooperate on a task without getting ahead

ways.

It would have units of

or behind. Adding two numbers might require, say, 30 clock cycles.

distance. 2. The amplitude of a

Microcomputers these days operate at clock frequencies of about a

swinging pendulum would more

gigahertz.

naturally be defined as an angle.

We have discussed how to measure how fast something vibrates,

but not how big the vibrations are. The general term for this is

162

Chapter 8

Waves

amplitude, A. The definition of amplitude depends on the system

being discussed, and two people discussing the same system may

not even use the same definition. In the example of the block on

the end of the spring, 8.1/1, the amplitude will be measured in

distance units such as cm. One could work in terms of the distance

traveled by the block from the extreme left to the extreme right, but

it would be somewhat more common in physics to use the distance

from the center to one extreme. The former is usually referred to as

the peak-to-peak amplitude, since the extremes of the motion look

like mountain peaks or upside-down mountain peaks on a graph of

position versus time.

In other situations we would not even use the same units for am-

plitude. The amplitude of a child on a swing, or a pendulum, 8.1/2,

would most conveniently be measured as an angle, not a distance,

since her feet will move a greater distance than her head. The elec-

trical vibrations in a radio receiver would be measured in electrical

units such as volts or amperes.

In many physical examples of vibrations, the force that brings

the vibrating object back to equilibrium gets stronger and stronger

as the object gets father and farther from equilibrium, and the force

is directly proportional to the distance from equilibrium.

Most

springs behave this way, for example, so for example we’d expect

that the spring in figure 8 would make very nearly twice the force

when stretched twice as much. We then define a spring constant,,

which tells us how many newtons of force we get per meter of stretch-

ing. For example, the John Hancock Tower has a spring constant of

about 200 MN/m (meganewtons per meter), meaning that the wind

must exert a force of about 200 MN in order to make the tower sway

by one meter. To make it sway by two meters, the force would have

to be 400 MN.

When the force has this type of mathematical behavior, the re-

sulting motion is known as simple harmonic motion. One surprising

and useful fact about simple harmonic motion is that its frequency

is independent of amplitude. Intuitively, we would expect that vi-

brations with a greater amplitude would take more time, i.e., have a

c / The

John

Hancock

Tower

lower frequency. However, when the amplitude is greater, the force

in Boston vibrates naturally at a

frequency of 0.14 Hz.

Surpris-

accelerating the mass back toward the equilibrium position is also

ingly, this frequency is the same

greater, and this turns out to compensate exactly for the need to

regardless of the amplitude of the

travel a greater distance. Legend has it that Galileo first noticed

vibrations

this fact when he watched a chandelier swinging during a church

service, and timed it against his pulse. Mathematically, the fre-

quency of vibration is given by f = (1/2π)pk/m, where k is the

spring constant, and m is the mass that is vibrating.

Section 8.1

Vibrations

163

d / The two circular patterns of

ripples pass through each other.

Unlike material objects, wave pat-

terns can overlap in space, and

when this happens they combine

by addition.

8.2 Wave motion

There are three main ways in which wave motion differs from the

motion of objects made of matter.

1. Superposition

The first, and most profound, difference between wave motion

and the motion of objects is that waves do not display any repulsion

of each other analogous to the normal forces between objects that

come in contact. Two wave patterns can therefore overlap in the

same region of space, as shown in the figure at the top of the page.

Where the two waves coincide, they add together. For instance,

suppose that at a certain location in at a certain moment in time,

each wave would have had a crest 3 cm above the normal water

level. The waves combine at this point to make a 6-cm crest. We

use negative numbers to represent depressions in the water. If both

waves would have had a troughs measuring -3 cm, then they combine

to make an extra-deep -6 cm trough. A +3 cm crest and a -3 cm

trough result in a height of zero, i.e., the waves momentarily cancel

each other out at that point. This additive rule is referred to as

the principle of superposition, “superposition” being merely a fancy

word for “adding.”

Superposition can occur not just with sinusoidal waves like the

ones in the figure above but with waves of any shape. The figures

on the following page show superposition of wave pulses. A pulse is

simply a wave of very short duration. These pulses consist only of

164

Chapter 8

Waves

a single hump or trough. If you hit a clothesline sharply, you will

observe pulses heading off in both directions. This is analogous to

the way ripples spread out in all directions when you make a distur-

bance at one point on water. The same occurs when the hammer

on a piano comes up and hits a string.

Discussion question

A

In figure e, the fifth frame shows the spring just about perfectly

flat. If the two pulses have essentially canceled each other out perfectly,

then why does the motion pick up again? Why doesn’t the spring just stay

flat?

e / These pictures show the motion of wave pulses along a spring.

To make a pulse, one end of the

spring was shaken by hand. Movies were filmed, and a series of frame chosen to show the motion. 1. A pulse travels to the left. 2. Superposition of two colliding positive pulses. 3. Superposition of two colliding pulses, one positive and one negative.

Section 8.2

Wave motion

165

f / As the wave pattern passes the rubber duck, the duck stays put.

The water isn’t moving forward with the wave.

2. The medium is not transported with the wave.

Figure f shows a series of water waves before it has reached a

rubber duck (left), having just passed the duck (middle) and having

progressed about a meter beyond the duck (right). The duck bobs

around its initial position, but is not carried along with the wave.

This shows that the water itself does not flow outward with the

wave. If it did, we could empty one end of a swimming pool simply

by kicking up waves! We must distinguish between the motion of

the medium (water in this case) and the motion of the wave pattern

through the medium. The medium vibrates; the wave progresses

through space.

self-check A

In figure g, you can detect the side-to-side motion of the spring because

the spring appears blurry. At a certain instant, represented by a single

photo, how would you describe the motion of the different parts of the

spring? Other than the flat parts, do any parts of the spring have zero

velocity?

. Answer, p. 180

A worm

example 2

The worm in the figure is moving to the right. The wave pattern,

a pulse consisting of a compressed area of its body, moves to

the left. In other words, the motion of the wave pattern is in the

opposite direction compared to the motion of the medium.

g / As

the

wave

pulse

goes

by, the ribbon tied to the spring

is not carried along. The motion

of the wave pattern is to the

right, but the medium (spring) is

moving up and down, not to the

right.

166

Chapter 8

Waves

Surfing

example 3

The incorrect belief that the medium moves with the wave is often

reinforced by garbled secondhand knowledge of surfing. Anyone

who has actually surfed knows that the front of the board pushes

the water to the sides, creating a wake — the surfer can even

drag his hand through the water, as in in figure h. If the water was

moving along with the wave and the surfer, this wouldn’t happen.

The surfer is carried forward because forward is downhill, not be-

cause of any forward flow of the water. If the water was flowing

forward, then a person floating in the water up to her neck would

h / Example 3.

The surfer is

dragging his hand in the water.

be carried along just as quickly as someone on a surfboard. In

fact, it is even possible to surf down the back side of a wave, al-

though the ride wouldn’t last very long because the surfer and the

wave would quickly part company.

3. A wave’s velocity depends on the medium.

A material object can move with any velocity, and can be sped

up or slowed down by a force that increases or decreases its kinetic

energy. Not so with waves. The magnitude of a wave’s velocity

depends on the properties of the medium (and perhaps also on the

shape of the wave, for certain types of waves). Sound waves travel

at about 340 m/s in air, 1000 m/s in helium. If you kick up water

i / Example

4:

a

breaking

waves in a pool, you will find that kicking harder makes waves that

wave.

are taller (and therefore carry more energy), not faster. The sound

waves from an exploding stick of dynamite carry a lot of energy, but

are no faster than any other waves. In the following section we will

give an example of the physical relationship between the wave speed

and the properties of the medium.

Breaking waves

example 4

The velocity of water waves increases with depth. The crest of a

wave travels faster than the trough, and this can cause the wave

to break.

Once a wave is created, the only reason its speed will change is

if it enters a different medium or if the properties of the medium

change. It is not so surprising that a change in medium can slow

down a wave, but the reverse can also happen. A sound wave trav-

eling through a helium balloon will slow down when it emerges into

j / Example 5.

The boat has

the air, but if it enters another balloon it will speed back up again!

run up against a limit on its speed

Similarly, water waves travel more quickly over deeper water, so a

because it can’t climb over its

wave will slow down as it passes over an underwater ridge, but speed

own wave. Dolphins get around

up again as it emerges into deeper water.

the problem by leaping out of the

water.

Hull speed

example 5

The speeds of most boats, and of some surface-swimming ani-

mals, are limited by the fact that they make a wave due to their

motion through the water. The boat in figure j is going at the same

speed as its own waves, and can’t go any faster. No matter how

Section 8.2

Wave motion

167

hard the boat pushes against the water, it can’t make the wave

move ahead faster and get out of the way. The wave’s speed de-

pends only on the medium. Adding energy to the wave doesn’t

speed it up, it just increases its amplitude.

A water wave, unlike many other types of wave, has a speed that

depends on its shape: a broader wave moves faster. The shape

of the wave made by a boat tends to mold itself to the shape of

the boat’s hull, so a boat with a longer hull makes a broader wave

that moves faster. The maximum speed of a boat whose speed is

limited by this effect is therefore closely related to the length of its

hull, and the maximum speed is called the hull speed. Sailboats

designed for racing are not just long and skinny to make them

more streamlined — they are also long so that their hull speeds

will be high.

Wave patterns

If the magnitude of a wave’s velocity vector is preordained, what

about its direction? Waves spread out in all directions from every

point on the disturbance that created them. If the disturbance is

small, we may consider it as a single point, and in the case of water

k / Circular

and

linear

wave

waves the resulting wave pattern is the familiar circular ripple, k/1.

patterns.

If, on the other hand, we lay a pole on the surface of the water

and wiggle it up and down, we create a linear wave pattern, k/2.

For a three-dimensional wave such as a sound wave, the analogous

patterns would be spherical waves and plane waves, l.

Infinitely many patterns are possible, but linear or plane waves

are often the simplest to analyze, because the velocity vector is in

the same direction no matter what part of the wave we look at. Since

all the velocity vectors are parallel to one another, the problem is

effectively one-dimensional. Throughout this chapter and the next,

we will restrict ourselves mainly to wave motion in one dimension,

while not hesitating to broaden our horizons when it can be done

without too much complication.

Discussion questions

A

[see above]

B

Sketch two positive wave pulses on a string that are overlapping but

not right on top of each other, and draw their superposition. Do the same

for a positive pulse running into a negative pulse.

C

A traveling wave pulse is moving to the right on a string. Sketch the

velocity vectors of the various parts of the string. Now do the same for a

pulse moving to the left.

D

In a spherical sound wave spreading out from a point, how would

the energy of the wave fall off with distance?

l / Plane

and

spherical

wave

patterns.

8.3 Sound and light waves

168

Chapter 8

Waves

Sound waves

The phenomenon of sound is easily found to have all the char-

acteristics we expect from a wave phenomenon:

• Sound waves obey superposition. Sounds do not knock other

sounds out of the way when they collide, and we can hear more

than one sound at once if they both reach our ear simultane-

ously.

• The medium does not move with the sound. Even standing

in front of a titanic speaker playing earsplitting music, we do

not feel the slightest breeze.

• The velocity of sound depends on the medium. Sound travels

faster in helium than in air, and faster in water than in helium.

Putting more energy into the wave makes it more intense, not

faster. For example, you can easily detect an echo when you

clap your hands a short distance from a large, flat wall, and

the delay of the echo is no shorter for a louder clap.

Although not all waves have a speed that is independent of the

shape of the wave, and this property therefore is irrelevant to our

collection of evidence that sound is a wave phenomenon, sound does

nevertheless have this property. For instance, the music in a large

concert hall or stadium may take on the order of a second to reach

someone seated in the nosebleed section, but we do not notice or

care, because the delay is the same for every sound. Bass, drums,

and vocals all head outward from the stage at 340 m/s, regardless

of their differing wave shapes.

If sound has all the properties we expect from a wave, then what

type of wave is it? It must be a vibration of a physical medium such

as air, since the speed of sound is different in different media, such

as helium or water. Further evidence is that we don’t receive sound

signals that have come to our planet through outer space. The roars

and whooshes of Hollywood’s space ships are fun, but scientifically

wrong.1

We can also tell that sound waves consist of compressions and

expansions, rather than sideways vibrations like the shimmying of a

snake. Only compressional vibrations would be able to cause your

1Outer space is not a perfect vacuum, so it is possible for sounds waves to

travel through it. However, if we want to create a sound wave, we typically do

it by creating vibrations of a physical object, such as the sounding board of a

guitar, the reed of a saxophone, or a speaker cone. The lower the density of the

surrounding medium, the less efficiently the energy can be converted into sound

and carried away. An isolated tuning fork, left to vibrate in interstellar space,

would dissipate the energy of its vibration into internal heat at a rate many

orders of magnitude greater than the rate of sound emission into the nearly

perfect vacuum around it.

Section 8.3

Sound and light waves

169

eardrums to vibrate in and out. Even for a very loud sound, the

compression is extremely weak; the increase or decrease compared

to normal atmospheric pressure is no more than a part per million.

Our ears are apparently very sensitive receivers!

Light waves

Entirely similar observations lead us to believe that light is a

wave, although the concept of light as a wave had a long and tortu-

ous history. It is interesting to note that Isaac Newton very influen-

tially advocated a contrary idea about light. The belief that matter

was made of atoms was stylish at the time among radical thinkers

(although there was no experimental evidence for their existence),

and it seemed logical to Newton that light as well should be made of

tiny particles, which he called corpuscles (Latin for “small objects”).

Newton’s triumphs in the science of mechanics, i.e., the study of

matter, brought him such great prestige that nobody bothered to

question his incorrect theory of light for 150 years. One persua-

sive proof that light is a wave is that according to Newton’s theory,

two intersecting beams of light should experience at least some dis-

ruption because of collisions between their corpuscles. Even if the

corpuscles were extremely small, and collisions therefore very infre-

quent, at least some dimming should have been measurable. In fact,

very delicate experiments have shown that there is no dimming.

The wave theory of light was entirely successful up until the 20th

century, when it was discovered that not all the phenomena of light

could be explained with a pure wave theory. It is now believed that

both light and matter are made out of tiny chunks which have both

wave and particle properties. For now, we will content ourselves

with the wave theory of light, which is capable of explaining a great

many things, from cameras to rainbows.

If light is a wave, what is waving? What is the medium that

wiggles when a light wave goes by? It isn’t air. A vacuum is impen-

etrable to sound, but light from the stars travels happily through

zillions of miles of empty space. Light bulbs have no air inside them,

but that doesn’t prevent the light waves from leaving the filament.

For a long time, physicists assumed that there must be a mysterious

medium for light waves, and they called it the aether (not to be con-

fused with the chemical). Supposedly the aether existed everywhere

in space, and was immune to vacuum pumps. We now know that,

as discussed in chapter 6, light can instead be explained as a wave

pattern made up of electrical and magnetic fields.

8.4 Periodic waves

Period and frequency of a periodic wave

You choose a radio station by selecting a certain frequency. We

have already defined period and frequency for vibrations, but what

170

Chapter 8

Waves

do they signify in the case of a wave? We can recycle our previous

definition simply by stating it in terms of the vibrations that the

wave causes as it passes a receiving instrument at a certain point

in space. For a sound wave, this receiver could be an eardrum or

a microphone. If the vibrations of the eardrum repeat themselves

over and over, i.e., are periodic, then we describe the sound wave

m / A graph of pressure ver-

that caused them as periodic. Likewise we can define the period

sus time for a periodic sound

and frequency of a wave in terms of the period and frequency of

wave, the vowel “ah.”

the vibrations it causes. As another example, a periodic water wave

would be one that caused a rubber duck to bob in a periodic manner

as they passed by it.

The period of a sound wave correlates with our sensory impres-

sion of musical pitch. A high frequency (short period) is a high note.

The sounds that really define the musical notes of a song are only

the ones that are periodic. It is not possible to sing a non-periodic

sound like “sh” with a definite pitch.

n / A similar graph for a non-

The frequency of a light wave corresponds to color. Violet is the

periodic wave, “sh.”

high-frequency end of the rainbow, red the low-frequency end. A

color like brown that does not occur in a rainbow is not a periodic

light wave. Many phenomena that we do not normally think of as

light are actually just forms of light that are invisible because they

fall outside the range of frequencies our eyes can detect. Beyond the

red end of the visible rainbow, there are infrared and radio waves.

Past the violet end, we have ultraviolet, x-rays, and gamma rays.

Graphs of waves as a function of position

Some waves, like sound waves, are easy to study by placing a

detector at a certain location in space and studying the motion as

a function of time. The result is a graph whose horizontal axis is

time. With a water wave, on the other hand, it is simpler just to

look at the wave directly. This visual snapshot amounts to a graph

of the height of the water wave as a function of position. Any wave

o / A strip chart recorder.

can be represented in either way.

An easy way to visualize this is in terms of a strip chart recorder,

an obsolescing device consisting of a pen that wiggles back and forth

as a roll of paper is fed under it. It can be used to record a per-

son’s electrocardiogram, or seismic waves too small to be felt as a

noticeable earthquake but detectable by a seismometer. Taking the

seismometer as an example, the chart is essentially a record of the

ground’s wave motion as a function of time, but if the paper was set

to feed at the same velocity as the motion of an earthquake wave, it

would also be a full-scale representation of the profile of the actual

wave pattern itself. Assuming, as is usually the case, that the wave

velocity is a constant number regardless of the wave’s shape, know-

ing the wave motion as a function of time is equivalent to knowing

it as a function of position.

Section 8.4

Periodic waves

171

Wavelength

Any wave that is periodic will also display a repeating pattern

when graphed as a function of position. The distance spanned by

one repetition is referred to as one wavelength. The usual notation

for wavelength is λ, the Greek letter lambda. Wavelength is to space

p / A water wave profile cre-

as period is to time.

ated by a series of repeating

pulses.

q / Wavelengths of linear and circular water waves.

Wave velocity related to frequency and wavelength

Suppose that we create a repetitive disturbance by kicking the

surface of a swimming pool. We are essentially making a series of

wave pulses. The wavelength is simply the distance a pulse is able to

travel before we make the next pulse. The distance between pulses

is λ, and the time between pulses is the period, T , so the speed of

the wave is the distance divided by the time,

v = λ/T .

This important and useful relationship is more commonly writ-

ten in terms of the frequency,

v = f λ.

Wavelength of radio waves

example 6

. The speed of light is 3.0 × 108 m/s. What is the wavelength of

the radio waves emitted by KKJZ, a station whose frequency is

88.1 MHz?

172

Chapter 8

Waves

. Solving for wavelength, we have

λ = v /f

= (3.0 × 108 m/s)/(88.1 × 106 s−1)

= 3.4 m

The size of a radio antenna is closely related to the wavelength of

the waves it is intended to receive. The match need not be exact

(since after all one antenna can receive more than one wave-

length!), but the ordinary “whip” antenna such as a car’s is 1/4

of a wavelength. An antenna optimized to receive KKJZ’s signal

would have a length of 3.4 m/4 = 0.85 m.

r / Ultrasound, i.e., sound with fre-

quencies higher than the range

of human hearing, was used to

make this image of a fetus. The

resolution of the image is re-

lated to the wavelength, since

details smaller than about one

wavelength cannot be resolved.

High resolution therefore requires

a short wavelength, correspond-

ing to a high frequency.

The equation v = f λ defines a fixed relationship between any two

of the variables if the other is held fixed. The speed of radio waves

in air is almost exactly the same for all wavelengths and frequencies

(it is exactly the same if they are in a vacuum), so there is a fixed

relationship between their frequency and wavelength. Thus we can

say either “Are we on the same wavelength?” or “Are we on the

same frequency?”

A different example is the behavior of a wave that travels from

a region where the medium has one set of properties to an area

where the medium behaves differently. The frequency is now fixed,

because otherwise the two portions of the wave would otherwise

s / A

water

wave

traveling

get out of step, causing a kink or discontinuity at the boundary,

into a region with a different

which would be unphysical. (A more careful argument is that a

depth changes its wavelength.

kink or discontinuity would have infinite curvature, and waves tend

Section 8.4

Periodic waves

173

to flatten out their curvature. An infinite curvature would flatten out infinitely fast, i.e., it could never occur in the first place.) Since

the frequency must stay the same, any change in the velocity that

results from the new medium must cause a change in wavelength.

The velocity of water waves depends on the depth of the water,

so based on λ = v/f , we see that water waves that move into a

region of different depth must change their wavelength, as shown in

figure s. This effect can be observed when ocean waves come up to

the shore. If the deceleration of the wave pattern is sudden enough,

the tip of the wave can curl over, resulting in a breaking wave.

174

Chapter 8

Waves

Problems

Key

√

A computerized answer check is available online.

R

A problem that requires calculus.

?

A difficult problem.

1

Many single-celled organisms propel themselves through water

with long tails, which they wiggle back and forth. (The most obvious

example is the sperm cell.) The frequency of the tail’s vibration is

typically about 10-15 Hz. To what range of periods does this range

of frequencies correspond?

2

(a) Pendulum 2 has a string twice as long as pendulum 1. If

we define x as the distance traveled by the bob along a circle away

from the bottom, how does the k of pendulum 2 compare with the

k of pendulum 1? Give a numerical ratio. [Hint: the total force

on the bob is the same if the angles away from the bottom are the

same, but equal angles do not correspond to equal values of x.]

(b) Based on your answer from part (a), how does the period of pen-

dulum 2 compare with the period of pendulum 1? Give a numerical

ratio.

?

3

The following is a graph of the height of a water wave as a

function of position, at a certain moment in time.

Trace this graph onto another piece of paper, and then sketch below

it the corresponding graphs that would be obtained if

(a) the amplitude and frequency were doubled while the velocity

remained the same;

(b) the frequency and velocity were both doubled while the ampli-

tude remained unchanged;

(c) the wavelength and amplitude were reduced by a factor of three

while the velocity was doubled.

Explain all your answers. [Problem by Arnold Arons.]

4

(a) The graph shows the height of a water wave pulse as a

function of position. Draw a graph of height as a function of time

Problem 4.

for a specific point on the water. Assume the pulse is traveling to

the right.

(b) Repeat part a, but assume the pulse is traveling to the left.

(c) Now assume the original graph was of height as a function of

time, and draw a graph of height as a function of position, assuming

the pulse is traveling to the right.

Problems

175

(d) Repeat part c, but assume the pulse is traveling to the left.

Explain all your answers. [Problem by Arnold Arons.]

5

Suggest a quantitative experiment to look for any deviation

from the principle of superposition for surface waves in water. Make

it simple and practical.

6

The musical note middle C has a frequency of 262 Hz. What

√

are its period and wavelength?

176

Chapter 8

Waves

Photo Credits

Except as specifically noted below or in a parenthetical credit in the caption of a figure, all the illustrations in this book are by under my own copyright, and are copyleft licensed under the same license as the rest of the book.

In some cases it’s clear from the date that the figure is public domain, but I don’t know the name of the artist or photographer; I would be grateful to anyone who could help me to give proper credit. I have assumed that images that come from U.S. government web pages are copyright-free, since products of federal agencies fall into the public domain. When “PSSC

Physics” is given as a credit, it indicates that the figure is from the second edition of the textbook entitled Physics, by the Physical Science Study Committee; these are used according to a blanket permission given in the later PSSC College Physics edition, which states on the copyright page that “The materials taken from the original and second editions and the Advanced Topics of PSSC PHYSICS included in this text will be available to all publishers for use in English after December 31, 1970, and in translations after December 31, 1975.”

In a few cases, I have made use of images under the fair use doctrine. However, I am not a lawyer, and the laws on fair use are vague, so you should not assume that it’s legal for you to use these images. In particular, fair use law may give you less leeway than it gives me, because I’m using the images for educational purposes, and giving the book away for free. Likewise, if the photo credit says “courtesy of ...,” that means the copyright owner gave me permission to use it, but that doesn’t mean you have permission to use it.

Cover Wave: Roger McLassus, GFDL 1.2.

Hand and photomontage: B. Crowell.

Table of Contents

Skateboarder:

Courtesy of J.D. Rogge, www.sonic.net/∼shawn.

Figure skater:

Wikimedia Commonus user Rosiemairieanne, GFDL 2.

Sunspot:

Royal

Swedish

Academy

of

Sciences.

The

astronomers’

web

page

at

http://www.solarphysics.kva.se/NatureNov2002/press images eng.html

states

“All

images

are free for publication.”.

X-Ray: 1896 image produced by Roentgen.

Surfing: Stan

Shebs, GFDL licensed (Wikimedia Commons).

7 Star trails: GFDL licensed, Wikipedia user Manfreeed.

8 Emmy Noether: Based on

Noether’s apparent age, the portrait must have been taken around 1900 or 1910, so it is in the public domain.

8 C.S. Wu: Smithsonian Institution, believed to be public domain.

8

Swan Lake: Peter Gerstbach, GFDL 1.2.

9 Portrait of Monsieur Lavoisier and His Wife:

Jacques-Louis David, 1788.

10 Astronaut: NASA.

15 Hockey puck: photo from Wikimedia

Commons, user Robludwig, GFDL/CC-BY-SA.

15 Portrait of James Joule: contemporary.

16 Aristotle: Francesco Hayez, 1811.

16 Jets over New York: U.S. Air Force, Tech. Sgt.

Sean Mateo White, public domain work of the U.S. Government.

17 Galileo’s trial: Cristiano

Banti (1857).

18 Rocket sled: U.S. Air Force, public domain work of the U.S. Govern-

ment.

17 Foucault and pendulum: contemporary, ca. 1851.

20 Skateboarder: Courtesy

of J.D. Rogge, www.sonic.net/∼shawn.

24 Welding: William M. Plate, Jr., public-domain

product of the U.S. Airforce, Wikimedia Commons.

24 Infrared photographs: Courtesy of

M. Vollmer and K.P. Möllmann, Univ.

Appl.

Sciences, Brandenburg, Germany, www.fh-

177

brandenburg.de/∼piweb/projekte/thermo galerie eng.html.

26 Newton: Godfrey Kneller,

1702.

30 Eclipse: 1919, public domain.

30 Newspaper headline: 1919, public domain.

34

Colliding balls: PSSC Physics.

53 Brahe: public domain.

57 Basebal pitch: Wikipedia user

Rick Dikeman, GFDL 1.2.

60 Bull: Photo by Wikimedia Commons user Bart Hiddink, CC-

BY.

61 Tornado: NOAA Photo Library, NOAA Central Library; OAR/ERL/National Severe

Storms Laboratory (NSSL); public-domain product of the U.S. government.

63 Longjump:

Thomas Eakins, public domain.

65 Pendulum: PSSC Physics.

64 Tetherball: Line art by

the author, based on a photo by The Chewonki Foundation (Flickr), CC-BY-SA 2.0 licensed.

83 Colliding nuclei: courtesy of RHIC.

84 Machine gunner’s body: Redrawn from a public-

domain photo by Cpl. Sheila Brooks.

84 Machine gunner’s head: Redrawn from a sketch by

Wenceslas Hollar, 17th century.

93 Lightning: C. Clark/NOAA photo library, public domain.

98 Ampère: Millikan and Gale, 1920.

102 Volta: Millikan and Gale, 1920.

104 Ohm: Mil-

likan and Gale, 1920.

111 Sunspot: Royal Swedish Academy of Sciences. The astronomers’

web page at http://www.solarphysics.kva.se/NatureNov2002/press images eng.html states “All images are free for publication.”.

122 Faraday banknote: fair use.

125 Maxwell: 19th

century photograph.

131 Rays of sunlight: Wikipedia user PiccoloNamek, GFDL 1.2.

134

Jupiter and Io: NASA/JPL/University of Arizona.

145 Ray-traced image: Gilles Tran, Wiki-

media Commons, public domain.

152 Flower: Based on a photo by Wikimedia Commons

user Fir0002, GFDL 1.2.

151 Moon: Wikimedia commons image.

161 Painting of waves:

Katsushika Hokusai (1760-1849), public domain.

163 John Hancock Tower: Wikipedia user

Sfoskett, GFDL 1.2.

165 Superposition of pulses: Photo from PSSC Physics.

166 Marker

on spring as pulse passes by: PSSC Physics.

167 Surfing (hand drag): Stan Shebs, GFDL

licensed (Wikimedia Commons).

167 Breaking wave: Ole Kils, olekils at web.de, GFDL li-

censed (Wikipedia).

172 Wavelengths of circular and linear waves: PSSC Physics.

173

Fetus: Image of the author’s daughter.

173 Changing wavelength: PSSC Physics.

178

Photo Credits

Hints and Solutions

Answers to Self-Checks

Answers to Self-Checks for Chapter 1

Page 9, self-check A: A conservation law in physics says that the total amount always remains the same. You can’t get rid of it even if you want to.

Page 13, self-check B: Exponents have to do with multiplication, not addition. The first line should be 100 times longer than the second, not just twice as long.

Page 26, self-check C: Doubling d makes d2 four times bigger, so the gravitational field experienced by Mars is four times weaker.

Answers to Self-Checks for Chapter 2

Page 40, self-check A: No, it doesn’t violate symmetry. Space-translation symmetry only says that space itself has the same properties everywhere. It doesn’t say that all regions of space have the same stuff in them. The experiment on the earth comes out a certain way because that region of space has a planet in it. The experiment on the moon comes out different because that region of space has the moon in it. of the apparatus, which you forgot to take with you.

Page 42, self-check B: The camera is moving at half the speed at which the light ball is initially moving. After the collision, it keeps on moving at the same speed — your five x’s all line on a straight line. Since the camera moves in a straight line with constant speed, it is showing an inertial frame of reference.

Page 43, self-check C: The table looks like this:

velocity (meters per second)

before the colli-

after the collision

change

sion

−1

0

+1

0

−1

−1

Observers in all three frames agree on the changes in velocity, even though they disagree on the velocities themselves.

Page 50, self-check D: The motion would be the same. The force on the ball would be 20

newtons, so with each second it would gain 20 units of momentum. But 20 units of momentum for a 2-kilogram ball is still just 10 m/s of velocity.

Answers to Self-Checks for Chapter 3

Page 65, self-check A: The definition of torque is important, and so is the equation F = ±F r.

The two equations in between are just steps in a derivation of F = ±F r.

Answers to Self-Checks for Chapter 4

Page 86, self-check A: The total momentum is zero before the collision. After the collision, the two momenta have reversed their directions, but they still cancel. Neither object has changed its kinetic energy, so the total energy before and after the collision is also the same.

179

Answers to Self-Checks for Chapter 5

Page 97, self-check A: Either type can be involved in either an attraction or a repulsion. A positive charge could be involved in either an attraction (with a negative charge) or a repulsion (with another positive), and a negative could participate in either an attraction (with a positive) or a repulsion (with a negative).

Page 97, self-check B: It wouldn’t make any difference. The roles of the positive and negative charges in the paper would be reversed, but there would still be a net attraction.

Answers to Self-Checks for Chapter 6

Page 124, self-check A: An induced electric field can only be created by a changing magnetic field. Nothing is changing if your car is just sitting there. A point on the coil won’t experience a changing magnetic field unless the coil is already spinning, i.e., the engine has already turned over.

Answers to Self-Checks for Chapter 7

Page 138, self-check A: Only 1 is correct. If you draw the normal that bisects the solid ray, it also bisects the dashed ray.

Page 141, self-check B: He’s five times farther away than she is, so the light he sees is 1/25

the brightness.

Page 148, self-check C: You should have found from your ray diagram that an image is still formed, and it has simply moved down the same distance as the real face. However, this new image would only be visible from high up, and the person can no longer see his own image.

Page 150, self-check D: Increasing the distance from the face to the mirror has decreased the distance from the image to the mirror. This is the opposite of what happened with the virtual image.

Answers to Self-Checks for Chapter 8

Page 166, self-check A: The leading edge is moving up, the trailing edge is moving down, and the top of the hump is motionless for one instant.

Solutions to Selected Homework Problems

Solutions for chapter 1

Page 33, problem 1:

10−3 g

10−3 kg

134 mg ×

×

= 1.34 × 10−4 kg

1 mg

1 g

Solutions for chapter 3

Page 68, problem 4: The pliers are not moving, so their angular momentum remains constant at zero, and the total torque on them must be zero. Not only that, but each half of the pliers must have zero total torque on it. This tells us that the magnitude of the torque at one end must be the same as that at the other end. The distance from the axis to the nut is about 2.5

cm, and the distance from the axis to the centers of the palm and fingers are about 8 cm. The angles are close enough to 90◦ that we can pretend they’re 90 degrees, considering the rough nature of the other assumptions and measurements. The result is (300 N)(2.5 cm) = (F )(8 cm), 180

or F = 90 N.

181

Index

absorption, 135

for relativistic momentum, 87

alchemy, 94

for time dilation, 72

ammeter, 101

coulomb (unit), 96

ampere (unit), 98

Coulomb’s law, 97

amplitude

current

defined, 163

defined, 98

peak-to-peak, 163

diffuse reflection, 136

angular magnification, 151

angular momentum, 62

Einstein, Albert, 29

introduction to, 61

electric current

Aristotle, 16, 25

defined, 98

aurora, 121

electric field, 115

electric forces, 95

black hole, 30

electrical energy, 23

Brahe, Tycho, 53

electromagnetism, 118

spectrum, 126

Catholic Church, 25

waves, 125

causality, 72

Empedocles of Acragas, 132

centi- (metric prefix), 11

energy

charge, 95

electrical, 23

conservation of, 97

equivalence to mass, 29, 87

Church

gravitational, 20

Catholic, 25

heat, 22

circuit, 100

kinetic, 20

complete, 100

Noether’s theorem, 27

open, 101

nuclear, 23

parallel, 108

sound, 22

series, 108

ether, 81

short, 107

circular motion, 52

Faraday, Michael, 122

complete circuit, 100

Fermat’s principle, 146

conductor

Feynman, Richard, 128

defined, 105

field, 111

conservation

electric, 115

of charge, 97

gravitational, 20, 113

of mass, 9

Newton’s law of gravity, 25

of momentum, 40

magnetic, 119

conservation law, 7

force

converging, 148

definition, 47

conversions of units, 13

of gravity, 49

Copernicus, 18

pairs, 47

correspondence principle

unit, 47

defined, 31

Foucault, Léon, 17

for mass-energy equivalence, 31

frame of reference, 17

182

inertial, 17

elliptical orbit law, 54

noninertial, 17, 53

equal-area law, 54

Franklin, Benjamin

law of periods, 54

definition of signs of charge, 96

Kepler, Johannes, 8, 53

free fall, 25

Keynes, John Maynard, 94

French Revolution, 11

kilo- (metric prefix), 11

frequency, 161

kinetic energy, 20

defined, 162

compared to momentum, 45

Galileo, 133

Lavoisier

projectile motion, 50

Antoine-Laurent, 9

Galileo Galilei, 16

least time, principle of, 146

garage paradox, 78

lever, 20

Gates, Bill, 28

light, 168

Gell-Mann, Murray, 37

absorption of, 135

generator, 123

as an electromagnetic wave, 126

gravitational constant, G, 27

particle model of, 136

gravitational energy, 20

ray model of, 136

gravitational field, 113

speed of, 133

Newton’s law of gravity, 25

wave model of, 136

gravity

Lorentz transformation, 75

force of, 49

Lorentz, Hendrik, 75

Newton’s law of, 25

magnetic field, 119

magnetism

heat, 22

and relativity, 116

Hertz, Heinrich, 126

caused by moving charges, 116

Hipparchus, 143

magnetic field, 119

Hooke, 94

related to electricity, 118

Hugo, Victor, 93

magnification

images

angular, 151

formed by curved mirrors, 148

by a converging mirror, 149

formed by plane mirrors, 147

Mars

of images, 151

life on, 121

real, 149

mass

virtual, 147

conservation of, 9

induction, 122

equivalence to energy, 29, 87

inertia

mass-energy, 29

principle of, 39

Mathematical Principles of Natural Philoso-

inertia, principle of, 16

phy, 27

insulator

Maxwell, James Clerk, 125, 126

defined, 105

mega- (metric prefix), 11

inverse-square law, 26, 140

metric system

Io, 134

prefixes, 11

Michelson-Morley experiment, 81

joule (unit), 20

micro- (metric prefix), 11

Jupiter, 134

milli- (metric prefix), 11

model, 7

Kepler’s laws, 54

momentum

Index

183

compared to kinetic energy, 45

ray model of light, 136

conservation of, 40

reflection

relativistic, 85

diffuse, 136

moon

specular, 138

distance to, 143

relativity

gravitational field experienced by, 27

and magnetism, 116

orbit, 51

resistance, 104

motion

retina, 150

periodic, 162

reversibility, 144

RHIC accelerator, 83

nano- (metric prefix), 11

Roemer, 133

Neanderthals, 67

rotational symmetry, 15

neutral (electrically), 97

newton (unit), 47

schematics, 107

Newton, Isaac, 93, 151

scientific notation, 12

apple myth, 27

sea-of-arrows representation, 114

law of gravity, 25

series circuit

third law, 47

defined, 108

Noether’s theorem, 8

short circuit

for angular momentum, 67

defined, 107

for energy, 27

sinks in fields, 114

for momentum, 43

Sokal, Alan, 31

Noether, Emmy, 8

sound, 168

nuclear energy, 23

energy, 22

speed of, 167

Oersted, Hans Christian, 115

sources of fields, 114

ohm (unit), 105

spring constant, 163

Ohm’s law, 105

sunspots, 121

ohmic

symmetry, 7

defined, 105

rotational, 15

open circuit, 101

time, 27

Optics, 27

translation, 38

Orion Nebula, 23

Système International, 11

parabola, 50

time reversal, 144

parallax, 142

time symmetry, 27

parallel circuit

torque

defined, 108

defined, 64

particle model of light, 136

transformer, 124

period

units, conversion of, 13

defined, 162

Principia Mathematica, 27

velocity

principle of superposition, 164

addition of, 19

projectile motion, 50

vision, 132

pulse

volt (unit)

defined, 164

defined, 102

Pythagoras, 132

voltage, 101

ray diagrams, 138

wave

184

Index

electromagnetic, 126

wave model of light, 136

wavelength, 126

work, 57

Wu, Chien-Shiung, 8

Index

185

Table of Contents

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 6

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 7

Chapter 8

Chapter 8

Chapter 8

Chapter 8

Chapter 8

Chapter 8

Chapter 8

Chapter 8

Chapter 8