5

Electromagnetic waves

I have also a paper afloat, with an electromagnetic theory of light,
which, till I am convinced to the contrary, I hold to be great guns.
James Clerk Maxwell, in a letter
to his friend, C.H. Cay, January 5, 1865.

5.1 The electric field

Let us recall the plan we adopted at the close of section 3.1:
(a) Learn about waves.
(b) Find that light does indeed have a wave-like character.
(c) Learn what the "something" is out of which light waves are formed.
In subsequent sections, we found that waves do have the properties of
reflection and refraction that light possesses. Then we shifted perspective
and found that light has the property of interference that waves possess. In
short, chapters 3 and 4 taught us that light does indeed have a wave-like
character.
Before we can address part (c) of the plan, we need to know certain
aspects of electricity and magnetism. So we turn now to those twin subjects.
Sometimes, when you pull a sweater off over your head on a day when
the air is particularly dry (perhaps indoors in winter), you hear a crackling
sound and may even feel an electric spark. The frictional rubbing of the
sweater on your shirt or blouse separates some electric charges, one from
another; the crackling and the spark are a miniature form of thunder and
lightning.
More deliberate rubbing of two different objects separates electric
charges more effectively. Specifically, if one rubs a glass rod with an old silk
handkerchief, electrons from the rod remain with the silk, leaving the rod
positively charged by default. Recall that an electron is a tiny negatively
charged particle. Its motion in a copper wire in your house constitutes an
electric current, and electrons form the outer portions of an atom. The
positive charges in an atom arise from the protons in the atomic nucleus. In
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5.1 THE ELECTRIC FIELD

Figure 5.1 Producing an electric force. Both the glass rod and the foil-covered ball are
positively charged (because each has fewer electrons than protons). The positive charges
are represented by circles with a plus sign inside.

the glass rod's initial neutral state, the number of electrons equalled the
number of protons. When some electrons are removed, the rod has more
protons than electrons and exhibits a net positive charge.
In this chapter and the next two, all electric charges arise either from
electrons (which have a negative charge) or from protons (which have a
positive charge). Those charges are equal in magnitude but opposite in sign.
Figure 5.1 shows a styrofoam ball, covered with aluminum foil and suspended by a silk thread. If we momentarily touch the metal foil with the
positively charged glass rod, some of the foil's electrons (which are attracted by the rod's positive charge) leave the foil, attach themselves to the rod,
and reduce the rod's net positive charge. Now the ball is positively charged
(by default), and the rod remains positively charged (at least somewhat so).
In short, we have two positively charged objects. And we find that now we
can push the ball around without touching it. Holding the positively charged
rod first on one side of the ball and then on the other, we can make the ball
swing to and fro, as though it were a child on a swing. Or we can make the
ball cruise in a horizontal circle, tethered by the silk thread. With the
charged rod, we are able to exert a force on the charged ball. (Recall that,
in physics, a force is a push or a pull.)
The preceding paragraph gives us the entire scene at once. Now we begin
to isolate pieces and describe them separately.
First we study the charged rod in isolation. Figure 5.2 shows the rod. Let
us imagine taking a separate positive charge, holding it still at a specific
point in space near the rod, and measuring the electric force exerted on it.
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5 ELECTROMAGNETIC WAVES

Figure 5.2 A positively charged rod and the electric field in its vicinity.

Then, at the corresponding location in the figure, we draw an arrow pointing in the direction of the force we measured. The length of the arrow is
drawn proportional to the magnitude of the force. After we have done this
for many locations - some close to the rod and others farther away - the
picture in figure 5.2 emerges.
Next, physics says that this set of arrows is a picture of something really
existing there in space. That something is called the electricfield.The word
"field" means that an arrow is associated with each point in space. Here is
an analogy. Imagine a wind-blown field of wheat on a gusty day. In some
places, the wheat stalks are aligned in a northward direction; in other
places, the stalks have been blown down toward the southeast, and so on.
At every location in the farmer's field, there is a wheat stalk, and it points in
some direction. Thus, in an analogy, each arrow in figure 5.2 could
represent a real physical wheat stalk. But the objective of thefigureis not to
describe wheat fields. Rather, each arrow represents the electric field at the
location of the arrow's tail. The next few paragraphs take us from this
beginning to a precise definition.
We generated figure 5.2 by measuring the force on an extra charge, and
that very process provides a definition of the electric field. In words alone,
the definition is this:
Value of electric field at point P =

force on charge we place at P
,
amount of charge we use

provided the charge we use is at rest at P. Why the division? If we were to
use a charge three times as large, we would find three times as much force.
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5.1 THE ELECTRIC FIELD

To get a quantity that is independent of how much charge we use, we need
to divide the force by the amount of charge.
Some symbols will be useful. A force has a direction as well as a
magnitude. When we need to display the directional aspect (as well as the
magnitude), we denote a force by the boldface letter F. The electric field at
a point also has a direction as well as a magnitude, and so its symbol is the
boldface letter E. For the amount of charge in the present context, we use
the symbol q. The symbol q provides also a name for a specific charge, when
we need to refer to the charge. (The next paragraph shows both uses of the
letter q: amount of charge and name.)
The verbal definition of the electric field given above becomes this:
F
Value of electric field at point P = E = - ^ (1)
q
when charge q is at rest at P. The subscript "ong" tells us that the force
Fon<? acts on charge q (and here the symbol q serves as a name for the
electric charge). In the denominator, the symbol q denotes the amount of
charge on "charge q." Figure 5.3 shows the definition in action.
The idea of the electric field can be an elusive concept, but at least our
definition has a notable merit: it is an operational definition. The adjective
"operational" means that the definition is couched in terms of acts - operations - that can be performed in the laboratory or elsewhere. While one
may have a hard time grasping what the electric field is, at least one can
determine its value at a point by explicit experiment.

Figure 5.3 Defining the electric field. Here the charge q has a value of 2 charge units.
Therefore E=(F o n q )/2, and the electric field arrow is half as long as the force arrow.

Conceptual framework

Now we return to figure 5.1: the charged rod and the charged ball. The ball
experiences a force. That is the obvious fact. Physics, however, conceives of
that force as arising in a two-step process.
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5 ELECTROMAGNETIC WAVES

(1) The first charge (on the rod) alters the real physical properties of
space everywhere: it sets up an "electric field" in space. To be
sure, the first charge's influence on the properties of space
diminishes with distance; yet the first charge "adds" something to
otherwise empty space.
(2) The second charge (on the foil-covered ball) is acted on by the real
electric field - the alteration of the properties of space - in its
immediate vicinity. Thus the second charge is acted on only by its
immediate surroundings.
This two-step conception elevates the electric field to the status of something real, something existing in space.
Why the field concept?

Why should physics want to introduce the idea of an electric field as an
element of physical reality?
In response, let us consider radio transmission between the earth and a
planetary probe, as sketched in figure 5.4. It takes about 30 minutes for a
radio message to go from the earth to the vicinity of Jupiter. Radio waves
are produced by electric charges moving in an antenna. We can wiggle the
charges, get the message underway, and then hold the charges at rest in
such a way that the antenna is electrically neutral. The radio message keeps
on going. There must be something real out there in inter-planetary space,
something that bears the message. Part of the something is a non-zero
electric field. The field concept provides a real physical carrier for the radio
message, a "something" that travels through what is otherwise empty
space.

Figure 5.4 A radio message en route from the earth to the vicinity of Jupiter. The scene is
set 15 minutes after a short message has been sent. The antenna on earth and the
planetary probe are visible objects - but there must be something else, too.

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5.2 FRANKLIN'S CLAPPER

5.2 Franklin's clapper

A more mundane example of an electric field is in order, too. Imagine two
thin brass disks, their diameters being about the length of your hand. As
figure 5.5 indicates, the disks are arranged to face each other and are
separated by a few centimeters. The left-hand disk is charged positively; the
right-hand disk, negatively. With such charges around, there must be an
electric field. In which direction does it point?

Figure 5.5 Franklin's clapper. The brass disks are seen edge-on; they are held up by posts
that do not conduct electricity (and that are omitted in the sketch). Positive and negative
charges are represented by circles with a plus and a minus sign inside, respectively. The
iron nut, hung from a silk thread, is pushed to and fro between the charged disks. (A
Wimshurst machine will charge up the disks nicely.)

Suppose we take an extra positive charge q and place it somewhere
between the two disks. The positive charges on the left-hand disk will repel
q toward the right. The negative charges on the right-hand disk will attract q
to the right. The two forces add to give a bigger force to the right. The
definition of the electric field E,
(i)

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5 ELECTROMAGNETIC WAVES

implies that E points to the right. This orientation will hold everywhere in
the space between the two disks.
Now we dangle an iron nut, suspended by a silk thread, into the central
region. Initially, the nut is electrically neutral, but we can charge it by
letting it touch the left-hand disk. Some electrons are attracted from the nut
to the disk, and the nut becomes positive by default.
We can use the electric field concept to figure out the force that now acts
on the iron nut. If we multiply equation (1) above by q on both sides and
then cancel on the right-hand side, we find
<7E = F o n ? .

(2)

The significance of the new equation is this: once we know the electric field
E, we can compute the force on any charge q by multiplying E by the
amount of charge q. At the moment, q is positive; E points to the right; and
so the force on the iron nut points to the right. The nut is pushed over to the
right-hand disk.
When the nut arrives, however, electrons swarm from the disk, neutralize
the once-positive nut, and then overdo things, making the nut negatively
charged. In equation (2), the amount of charge q is now a negative quantity;
the electric field E still points to the right; but the interpretation of the
product qE (with q negative) is that the force points in the opposite direction: toward the left. The nut is shoved back to where it started.
When the nut arrives at the left-hand disk, electrons will leave the nut,
and the nut will become positive once more. The stage is set for another
swing to the right and then back. And on and on, so long as the disks remain
highly charged.
Although one still cannot literally see the electric field, one can hear its
effect: each time the iron nut swings over to a disk, it hits with a tiny
"clink." Benjamin Franklin invented this simple noise maker, and it carries
the name "Franklin's clapper."

5.3 The magnetic field
Most of us are familiar with magnetism, at least in an informal way. Cartoons and shopping lists decorate the refrigerator door, held there by little
magnets. A compass needle tells the hiker the direction of north. More
precisely, the needle tells the direction of the magnetic north pole, located
in the Canadian high arctic, but south of the rotational north pole. Figure
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5.3 THE M A G N E T I C FIELD

Figure 5.6 How a compass needle points at various locations on the earth's surface. Two
poles are shown: the true north pole, about which the earth rotates, and the magnetic
north pole, whose location is marked by the triangle. The short arrows indicate how a
compass needle points; they become a map of the earth's magnetic field.

5.6 sketches a portion of the earth's surface and shows how a compass
needle points at various locations.
At each location on the earth's surface, the earth's magnetism orients a
compass needle in a specific direction. The needle swings but settles down
to a specific direction, reproducibly. From the perspective of someone
above the earth and looking at it, the needle direction varies greatly from
Alaska to Florida but does so smoothly.
In short, the earth's magnetic effect on a compass needle varies from
place to place, and so physics introduces the idea of a magnetic field.
The direction of the magnetic field is specified operationally to be the
direction in which a compass needle lines up. Certainly this definition serves
to determine the direction of the earth's magnetic field, and it suffices (for
our purposes) in other contexts, too. Moreover, a compass needle can be
our detector of magnetic fields.
The symbol for the magnetic field is the boldface letter B. This is the
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5 ELECTROMAGNETIC WAVES

magnetic analog of E for the electric field. (Unfortunately, the symbol B
provides no mnemonic aid. The choice of symbol is an alphabetical accident
of history. When James Clerk Maxwell introduced a comprehensive set of
symbols for electricity and magnetism, the letter M had already been
claimed by another physical quantity. Because the magnetic field came
second on the list of quantities yet to be given symbols, it received the
second letter of the alphabet: B.)
The value of the electric field at a point has both a direction and a
magnitude. So, too, does the value of the magnetic field, but we have
defined the field's direction only. The reason for the omission is this: an
operational definition of magnitude is technically complex; we do not need
the magnitude; and so we are best off ignoring it. But some further experience with magnetism is certainly in order.
The Oersted experiment

A car battery can produce a steady flow of electrons through a copper wire
connected to its terminals (at least for intervals of a few seconds). The flow
of charge constitutes an electric current. Figure 5.7 sketches that context

Figure 5.7 The Oersted experiment: a steady electric current produces a magnetic field.
The compass needle points one way below the current-carrying wire and in the opposite
direction above the wire.
A note about the electric current. When the copper wire is connected to the battery,
electrons already in the wire - electrons that come from the copper atoms - begin to
flow through the wire. As some of those electrons enter the battery through one
terminal, other electrons emerge from the battery at the other terminal and enter the
wire. The wire started off electrically neutral (but containing both electrons and positive
copper nuclei); it remains electrically neutral. Thus the deflection of the compass needle is
not a consequence of electric forces. Rather, because electrons move through the wire,
there is a flow of electric charge and hence an electric current. That current is responsible
for the deflection of the compass needle, and we infer the existence of a magnetic field as
a real, physical intermediary.

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5.3 T H E M A G N E T I C FIELD

and also shows how a compass needle responds. When placed above a
straight section of the current-carrying wire, the needle points one way;
when placed directly below, the needle swings around and points the other
way. If only the earth's magnetic field were present, the needle would point
the same way in both locations. Here we have clear evidence that electric
charges in steady motion - a steady electric current - produce a magnetic
field.
By 1820, the Danish physicist Hans Christian Oersted had long been
looking for a connection between electricity and magnetism. While preparing one of his lectures, he thought of an experimental arrangement of
electric current and compass needle that might reveal a connection. He
assembled the apparatus but did not have time to try it out before class. As
he lectured, his conviction grew - yes, the experiment would show a connection - and so Oersted tried the experiment right there in class. The
compass needle responded to the electric current, and so the experiment
sketched in this section carries the name "the Oersted experiment." It was
the first experiment to establish a connection between electricity and
magnetism. (Professor Oersted had only a weak source of electric current,
and so the needle's response was feeble, indeed, barely perceptible. In his
words, "the experiment made no strong impression on the audience."
Moral: regardless of whether they are innovative or shopworn, lecture
demonstrations need to be large, visible, and convincing.)
Magnetic fields produced by electric currents are all around us, but we
are usually unaware of them. For example, the earth's magnetic field is
believed to be produced by electric currents deep in the earth's molten
interior. Scrap metal and old cars are picked up in a junk yard with an
electromagnet: a magnet produced by an electric current. To drop the junk,
the crane operator just turns off the current.

Magnetic forces
An electric current, we have learned, produces a magnetic field. Does a
reciprocal effect occur, that is, does a magnetic field affect an electric
current?
The answer is yes. Perhaps you have sat in front of a computer terminal
and watched the letters and numerals appear on the screen. You may know
that an electron beam (inside the TV-like tube) produces the light by impact
on the inside of the tube face. But what steers the beam around from one
letter to another? Or through the pen strokes of the letter A? Two electromagnets, buried deep inside the tube, perform those tasks. Their magnetic
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5 ELECTROMAGNETIC WAVES

fields push on the electrons and deflect the beam, steering it from place to
place on the tube face.
5.4 The entities
A pause to summarize is in order. What are the entities that we have met in
this chapter?
• Electric charge. Electrons and protons are tiny particles of matter.
The electron has a negative charge, and the proton, a positive
charge. (The hydrogen atom - the simplest of all atoms - consists
of merely an electron and a proton. The two charged particles
attract each other and so stay near each other, forming an atom.)
• Electric current. An electric current is nothing but electric charge in
motion. The flow of electrons in a copper or aluminum wire is
what carries "electricity" around your house.
• The electric field E.
• The magnetic field B.
But what, you may ask, are the electric and magnetic fields? They are
defined operationally in terms of forces on charged particles or compass
needles. They can exist in what is otherwise vacuum (as in the instance of a
radio message sent to a planetary probe near Jupiter). The fields are
fundamental; I cannot express them in terms of anything more primitive or
more intuitive. We can visualize the pattern they make in space, but we
cannot "see" them literally.
In our conceptual framework, the fields are the agents immediately
responsible for electric and magnetic forces. For example, the foil-covered
ball in figure 5.1 responds to the electric field in its vicinity and experiences
an electric force. To be sure, the charged rod produces the electric field, but
physics insists on the electric field as an intermediary between the charged
rod and the charged ball.
5.5 Faraday's law
We just noted that electric and magnetic fields exert forces on charged
particles. Physics takes the field idea very seriously, and so we are led to
ask, can one kind of field have an effect on the other field? Specifically, can
one kind of field produce the other? (In the contexts that we have studied
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5.5 FARADAY'S LAW

Figure 5.8 Moving the bar magnet relative to the coil of wire causes the needle of the
current meter to deflect. Inference: when a magnetic field changes with time, an electric
field is produced.

thus far, the fields have been produced by electric charges, at rest or in
motion.)
Figure 5.8 sketches apparatus with which we ask, experimentally, can a
magnetic field produce an electric field? A coil of wire is connected to a
sensitive current meter. At the start, no current flows, and the meter reads
zero. Only if an electric field arises within the wire will current flow; then
the electrons will be pushed around the circuit consisting of wire and meter.
Thus a non-zero meter reading indicates the presence of an electric field.
A bar magnet provides a convenient source of magnetic field. The
magnet produces a strong field in its vicinity and not much field far away.
When we thrust the bar magnet into the coil of wires, the current meter
responds, indicating a flow of electrons. We infer that an electric field has
arisen. But the current quickly drops back to zero. The magnet is sitting
inside the coil; there is plenty of magnetic field inside and around the coil;
but there is no longer a current and hence no longer an electric field.
We remove the magnet from the coil - and again the meter responds, but
only momentarily.
A current arises when the magnet is being inserted or withdrawn, but
only then. During those periods, an electric field is produced. And during
those periods, the magnetic field at the location of the coil is changing,
either from zero to some large value or from the large value back to zero.
Our inference is this: a changing magnetic field generates an electric field.
A host of other experiments - some of which you may see - confirm this
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5 ELECTROMAGNETIC WAVES

conclusion. The name for the conclusion, Faraday's law, honors the English
physicist Michael Faraday, who discovered the effect in the 1830s. The son
of a blacksmith and originally apprenticed as a bookbinder, Faraday rose by
his industry and skill to become director of London's Royal Institution.
5.6 Electromagnetic waves

In the last section, we learned that a changing magnetic field generates an
electric field. Is there reciprocity? Does a changing electric field generate a
magnetic field?
The answer is yes, but here a simple experiment is not possible. Rather,
we need to do some reasoning and approach the question indirectly.
Here is the logic, expressed tentatively.
Because a changing magnetic field generates an electric field and
if a changing electric field generates a magnetic field,
then we may be able to have both fields changing and generating each
other. To be sure, we would need some electric charges to produce the
fields in the first place, but once we got the show going, the fields would
sustain themselves.
To peek ahead, if this conjecture is borne out, we can understand radio
transmission to the Jupiter probe.
Figure 5.9 shows a device for producing self-sustaining electric and

Expect electric
and magnetic
fields out here

To electrical
socket
Figure 5.9 An antenna for producing self-sustaining electric and magnetic fields. Electrons
are sent up and down the vertical wires periodically.

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5.6 ELECTROMAGNETIC WAVES

Figure 5.10 How the electric and magnetic fields appear. For the sake of our spatial
orientation, the antenna wires are shown on the far left, but the fields close to the wires
have been omitted from the sketch. The field values are given solely for locations along
the line extending rightward from the antenna's center. Similar patterns of E and B exist
along other lines emanating from the antenna.

magnetic fields. Its essence consists of two vertical pieces of wire and
internal circuitry to send electrons up and down those wires. When the
electrons have been sent up, the upper wire is negatively charged and the
lower wire, positively charged, as the sketch shows. A moment later, the
electrons will be sent down; the lower wire will be negatively charged, and
the upper wire, positively charged. Then the electrons will be sent back to
the upper wire and so on, periodically.
The motion of the electrons constitutes a current. We can expect that
current to produce a magnetic field. When one wire is positively charged
and the other negatively, each wire will make an electric field, and those
electric fields will not cancel each other. So we can expect, typically, a nonzero electric field. While this is not the whole story for a periodicallyvarying distribution of electric charge, it certainly suggests that the device
will generate both electric and magnetic fields. Optimistically, we call it the
transmitting antenna.
Figure 5.10 gives a perspective drawing of the fields that we can expect.
Because the electrons are driven up and down the wires periodically, the
fields show a periodic pattern. The orientation of the magnetic field is
similar to what figure 5.7, the Oersted experiment, showed for the field
produced by a steady current: an orientation perpendicular to the wire. The
electric field is lined up parallel to the way the charges are separated on the
antenna.
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5 ELECTROMAGNETIC WAVES

If this pattern of fields moves through space (along the line indicated), a
stationary observer - his name is Bob - notes temporal variation. As the
pattern passes Bob, first the electric field points up; then it decreases and
soon points down; next it increases and again points up. Similar behavior,
but tipped by 90 degrees, holds for the magnetic field. Such changes in time
(at a fixed location) provide the "changing electric field" and the "changing
magnetic field" that generate the magnetic and electric fields, respectively,
and keep the show going even after the antenna has been unplugged.
But how much of this is speculation and how much real? The circuity in
the transmitting antenna can easily send the electrons up and down the
wires with a frequency of oscillation equal to 3x 109 round-trips per second.
At a distance of about 4 meters, one can detect the electric field by letting it
push electrons through another wire, this one connected to a current meter.
The presence of a current implies the existence of an electric field to urge
the electrons along. The combination of wire and meter is our detector of
the electric field.
What does one find?
First, when we place the detector out where figures 5.9 and 5.10 suggest
that an electric field exists, the electric field is indeed there.
Second, when a large sheet of aluminum, 1 millimeter thick, is placed
between the antenna and the detector, the metal reads zero current and
implies no electric field at the detector location. But figure 5.11 reveals what
is going on. If the aluminum sheet is tilted relative to the line from the
antenna to the old detector location, then at the place labeled "new location" the detector does find an electric field. (No field was present there
originally.) The pattern of electric and magnetic fields has been reflected.
Indeed, one can hold the detector fixed at the new location and rotate the
aluminum sheet back and forth a bit (about a vertical axis running through
the sheet), alternately causing the reflected pattern to hit the stationary
detector and to miss it.
Third, if one uses an aluminum sheet with two "slits," each a rectangle 5
centimeters by 15 centimeters, one has the analog of the double-slit experiment with light. With the antenna as source, the experiment is not easy
(because stray reflections from walls confuse the pattern behind the
aluminum sheet), but a double-slit interference pattern can be found.
Other experiments are possible. For example, a giant prism made of
paraffin wax produces refraction of the traveling pattern.
We may take these experiments as good evidence that electric and
magnetic fields are able to sustain each other. Once started, the fields move
through space on their own, propagating in a wave-like fashion.
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5.6 ELECTROMAGNETIC WAVES

New location

Aluminium
sheet

Old detector
location

Figure 5.11 A top view that illustrates reflection of the traveling pattern of electric and
magnetic fields. In this configuration of antenna and aluminum sheet, electric field is no
longer found at the "old detector location," but electric field is observed at the "new

location."

The traveling pattern in figure 5.10 is one example of an electromagnetic
wave. The direction in which the pattern moves is called the propagation
direction. Both the electric field and the magnetic field are perpendicular to
the propagation direction. In chapter 3, we studied waves on a metal
skeleton; the motion of the ribs that form the wave pattern is perpendicular
to the motion of the wave crests and troughs. We called such waves
transverse waves. Electromagnetic waves are formed of electric and
magnetic fields that are perpendicular to the propagation direction; so it is
appropriate to say that electromagnetic waves are transverse waves, too.
James Clerk Maxwell

The hero of this chapter is James Clerk Maxwell. A Scottish physicist,
Maxwell studied Faraday's Experimental Researches and made himself master of all that was known about electricity and magnetism in the 1850s (the
decade before the American Civil War). He sought a single, unified description of these two broad classes of phenomena. Success came in 1864, and
Maxwell's paper, "A Dynamical Theory of the Electromagnetic Field,"
appeared in the Philosophical Transactions of the Royal Society in 1865.
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5 ELECTROMAGNETIC WAVES

Not only was there unification of the phenomena, but Maxwell's theory
predicted self-sustaining and propagating electric and magneticfields- what
we have called electromagnetic waves.
Maxwell was able to predict the speed at which such waves should travel:
3xlO 8 meters/second. The experimental data used in this prediction came
from a set of strictly electric and magnetic measurements on charges at rest
or in motion in wires. No experiments on light entered into the prediction.
As Maxwell remarked, 'The only use made of light in the experiment was
to see the instruments" and to read them. Yet we recognize the predicted
speed as coinciding with the speed of light, known since Newton's day.
Maxwell was well prepared to recognize the coincidence, too. A link
between light on the one hand and electricity and magnetism on the other
had been sought for decades, especially by Faraday, and with a modicum of
success. And so no one is surprised to find Maxwell writing as follows:
This velocity is so nearly that of light [when the comparison is
made to more decimal places] that it seems we have strong reason
to conclude that light itself (including radiant heat and other radiations, if any) is an electromagnetic disturbance in the form of
waves propagated through the electromagnetic field according to
electromagnetic laws.
No surprise, but the statement is staggering in its implications: light is an
electromagnetic wave.

5.7 The electromagnetic spectrum

To say that light is an electromagnetic wave means this: the red beam from a
neon laser consists of a traveling pattern of electric and magnetic fields. In
chapters 3 and 4, experiments told us that light is a wave phenomenon, but
those chapters left unanswered the question, what is the "something" out
of which light waves are formed? Now we have an answer: light waves are
formed out of electric and magnetic fields. The fields are the "something."
When light streaks invisibly from the laser to a red spot on the wall, the
intervening space contains electric and magnetic fields, a wave-like pattern
of them - indeed, a pattern traveling at the speed of light.
The wavelength of that pattern we already know: A,neonred=6.3xlCr7
meters. This is illustrated in figure 5.12. We can imagine an observer
stationed somewhere along the beam. The pattern travels by at the speed c;
so high a speed and so short a wavelength send crest after crest past the
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5.7 THE ELECTROMAGNETIC SPECTRUM

Laser

Figure 5.12 W h e n a neon laser sends a continous light beam t o the distant wall, electric
and magnetic fields fill an approximately cylindrical volume of space, the space occupied by
the beam. (The cylinder is about I millimeter in diameter.) An enlarged rendition of a tiny
section reveals the fields and illustrates the wavelength.

observer at an astonishingly high frequency. Our general relationship
among frequency, wavelength, and speed is

Specializing this equation to light, we have
(1)

To diplay the frequency/in isolation, we divide both sides by X:

f = -J

X

And so, for the neon beam, we find
f
-/neon red

-.

c

^neon red

_ 3xlO 8 meters / second
6.3 xlO" 7 meters
~ \ x 1015 = 5 x 1014 oscillations / second.

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5 ELECTROMAGNETIC WAVES

At a fixed location, the electric field changes from up to down and back to
up about 5xlO 14 times each second.
The transmission antenna of section 5.6 sent electrons up and down the
vertical wire at a frequency of /=3xlO 9 oscillations/second. The traveling
pattern is repeated at this frequency. What is the pattern's wavelength?
Now we need to isolate X, in equation (1); division on both sides by/does
the job:

f
And we find
,

3xlO 8

=0.1 meters.

9

3xl0
The antenna sends out waves with a wavelength of 10 centimeters, roughly
the width of your hand.
We have now two instances of electromagnetic waves: the 10-centimeter
waves and the neon laser beam. They are but two points along an entire
spectrum of possibilities. Nature provides us with electromagnetic waves
having other frequencies and wavelengths, and technology enables us to
produce some at will. Figure 5.13 shows major ranges in the electromagnetic
spectrum. Visible light occupies a narrow interval, bounded by infrared

10 4

10 6

10 8

10 10

10 12

10 14

10 16

10 18

Frequency
increasing

1
104
Wavelength
increasing

102

10°

10~2

10" 4

10~6

10~8

10- 10

Figure 5.13 The electromagnetic spectrum. The numerical values of frequency are given in
oscillations per second; those of wavelength, in meters. Nature provides a continuous
spectrum, but physicists have divided the spectrum up into intervals, some narrow but
others broad. All the types of radiation noted here share certain essential features: they
have a wave-like character, are formed from electric and magnetic fields, and travel (in
vacuum) at the speed of light.

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5.7 THE ELECTROMAGNETIC SPECTRUM

radiation (starting at frequencies below those of red light) and by ultraviolet
light (starting at frequencies higher than those of violet light). The 10centimeter waves of the last section are an instance of microwaves. The
electromagnetic waves that heat a croissant in a microwave oven have,
typically, a frequency of 2.45 xlO 9 oscillations/second and hence a
wavelength of 12.2 centimeters. If you roast a turkey at 350 degress Fahrenheit in a conventional oven, the radiation filling the oven is predominantly
infrared. (The skin of your hand is a good detector of that radiation, but the
retina of your eye is not.) If you tune in channel 3, your television set
searches for electromagnetic waves at a frequency of 61.25 x 106 oscillations/
second. The associated wavelength is X=c//=5 meters. Classical music from
an FM radio station at 88.6 FM is carried by electromagnetic waves whose
underlying frequency is 88.6xlO6 oscillations/second. Thus TV and FM
radio broadcast at similar frequencies. If, however, you are an aficionado of
old-fashioned AM radio, then 1080 on your dial brings in a message sent
from the broadcasting station on waves at a frequency of 1080xlO3 oscillations/second, a much lower frequency. The associated wavelength is huge:
about 300 meters. When police use radar to monitor the speed of highway
traffic, an electromagnetic wave travels from the cruiser, reflects off the
speeding car, and returns to an antenna on the cruiser. The wavelength of
such a radar beam is a few centimeters (anywhere from 1 to 5 centimeters,
really) and places the radiation in the microwave interval.
The frequencies above the visible are less common in our lives, but
certainly not absent. Light from the sun contains substantial ultraviolet light
(in addition to visible and infrared). A bad sunburn results from too much
ultraviolet light; you are especially vulnerable at the beach or on a ski slope,
where the surface reflects the ultraviolet back for a second chance at being
absorbed by your skin. When the dentist takes an X-ray of your molars,
electromagnetic waves with a frequency of perhaps 1.5xlO19 oscillations/
second pass through your jaw and teeth, producing a photograph on film
held in your mouth. Gamma rays are rare in our lives. Their origin is
typically the nuclei of atoms undergoing radioactive decay. Just as the
transmitting antenna of section 5.6 used the motion of electrons in wires to
generate microwaves, so the motion of protons in the atomic nucleus can
produce electromagnetic radiation, then called gamma radiation. (That
electromagnetic radiation was the third in a trio of once-mysterious kinds of
radiation, labeled alpha, beta, and gamma, merely the first three letters of
the Greek alphabet. The three kinds of radiation were lettered in order of
increasing ability to penetrate matter; the gamma rays were the most difficult to stop. The alpha radiation turned out to be just a compact cluster of
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5 ELECTROMAGNETIC WAVES

two protons and two neutrons; the beta type turned out to be ordinary
electrons; only the third kind - gamma radiation - is electromagnetic in
character.)

5.8 "Sidedness" explained and explored

Back in chapter 1, experiments with clip-on sunglasses or their equivalent
led us to ascribe some kind of "sidedness" to light. Now we can make
theoretical sense of this. In figure 5.10, the electromagnetic wave has the
electric field arrayed in one plane and the magnetic field in another plane,
perpendicular to the first. In colloquial language, the wave has electric field
pointing out on two sides (up and down in the figure) and magnetic field
pointing out on two other sides (in and out of the page). Thefieldsgive a
"sidedness" to an electromagnetic wave and hence to light. The "sides" of
the light wave that have E pointing out differ from the "sides" that have B
pointing out.
In what follows, we concentrate our attention on the electric field. The
magnetic field is always present, too, but working with one field is
sufficient.
First, what happens when light impinges on a sheet of Polaroid? Recall
that the sheet consists of stretched and partially-aligned strands of polyvinyl
alcohol, together with iodine atoms that string themselves along each
molecular chain. Figure 5.14 depicts the fate of light whose electric field is
either wholly parallel to the stretch direction or wholly perpendicular: the
light is either wholly absorbed or wholly transmitted. (No sheet does this
perfectly, but such is the ideal.) An analogy can help us to understand why
this happens.
The iodine-coated strands act somewhat like electrically conducting

E:

Transmission
axis

E:

Absorbed

or

Transmitted

Figure 5.14 The fate of a light wave impinging on a sheet of Polaroid. If the electric field is
parallel to the stretch direction, the wave is absorbed; if perpendicular, then transmitted.
The broad, double-headed arrow denotes the transmission axis.

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5.8 "SIDEDNESS" EXPLAINED AND EXPLORED

Transmitted

Figure 5.15 W h a t happens when the incident electric field is oblique to the transmission

axis.

wires. To be sure, not like very good conductors, but like conductors
nonetheless. If the electric field is parallel to the "wire," some electrons are
pushed along; there is microscopic current and heating (as when current
flows through the wires of an electric toaster). The energy for this heating
comes from the electromagnetic wave, and so the light is absorbed. If,
however, the electric field is perpendicular to the "wires," the field cannot
push the electrons far; there is no current, no heating, and no absorption.
Remarkably, figure 5.14 contains all the information about Polaroid
sheets that we need. If the electric field is oblique to the transmission axis,
as shown in figure 5.15, we decompose the field into components that are
wholly parallel and wholly perpendicular to the axis, as indicated. The
parallel component is transmitted; the perpendicular, absorbed. Some light
will emerge, but it will be dimmer in intensity. And, of course, the
emergent light will have its electric field purely along the direction of the
Polaroid's transmission axis.
The last sentence has several ramifications. Light whose electric field
always oscillates in the same plane is called plane polarized light. The
electromagnetic wave in figure 5.10 is plane polarized, for the electric field
always points either "up" or "down" in the plane of the page. The light
from a hot filament comes from many atoms and their individual electrons.
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5 ELECTROMAGNETIC WAVES

A

Figure 5.16 Making plane polarized light, (a) At a fixed point, the electric field of light
from a hot filament points first one way, then another, yet a third, and so on. The
sequence shows the field at different instants of time (but at the same location), (b) After
the light has passed through a Polaroid sheet, only the components parallel to the
transmission axis remain.
The lengths of the arrows vary, and so you might wonder whether our eyes would notice
some temporal variation in the intensity of the transmitted light. The answer is "no." The
changes follow one another so rapidly that our eyes respond only to an average over the
lengths of many arrows.

The ensuing electric field is a combination of many unrelated contributions.
Just what those contributions add up to varies significantly from moment to
moment. The field does not have a fixed plane of oscillation but rather
oscillates for a while this way, then that way, and so on. Figure 5.16
illustrates the situation. If we put a sheet of Polaroid in front of the filament, then only the component of E parallel to the transmission axis will be
allowed to pass. The perpendicular component is absorbed. The transmitted light is dimmer but plane polarized - and a good light source for an
experiment.
Another experiment

Indeed, let us put together a last experiment for this chapter. In front of a
frosted light bulb with its hot filament, we place a Polaroid sheet with the
transmission axis vertical. In front of that sheet we place another sheet, but
with its transmission axis horizontal. Between the two sheets is plenty of
light, its electric field oscillating in the vertical plane, but none of that light
gets through the last Polaroid sheet. Because the electric field of the "in
between" light is perpendicular to the last sheet's transmission axis, the last
sheet absorbs it all.
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5.8 "SIDEDNESS" EXPLAINED AND EXPLORED

Last
Figure 5.17 Stacking three Polaroid sheets. The last sheet has its transmission axis
perpendicular t o the first sheet's transmission axis. The middle sheet has its transmission
axis oriented obliquely relative to the axes of the other sheets. Can any light run this

gauntlet?

Nothing about that is surprising, but now suppose we insert a middle
Polaroid sheet between the first and last and do so at some oblique angle.
The transmission axis might be tilted 30 degrees relative to the first sheet's
transmission axis. Figure 5.17 provides a sketch. We have inserted another
absorber of light. But it is a selective absorber, absorbing only the component of electric field perpendicular to its transmission axis. Before we
jump to the obvious conclusion - that no light could possibly get through
this sequence of absorbers - let us work our way along, thinking about
components.
After light from the frosted bulb passes through the first sheet, the light's
electric field is purely vertical. When light reaches the middle sheet, as
sketched in part (a) of figure 5.18, the electric field component along that
sheet's transmission axis is transmitted. Thus the light traveling from the
middle sheet to the last sheet is plane polarized in an oblique direction.
When that light reaches the last sheet, its field component parallel to the
transmission axis will be transmitted. As the diagrams show, the portion
transmitted is not zero. In short, we predict that light will now emerge dimmer, certainly, than light coming directly from the bulb, but light
nonetheless.
The experiment is easy to set up - and it confirms the prediction. The
sidedness of light makes it possible for more absorbers to produce less
absorption!
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5 ELECTROMAGNETIC WAVES

Middle
transmission
Absorbed

Figure 5.18 Following the light through the sequence of Polaroid sheets, (a) When light
impinges on the middle sheet, (b) When light reaches the last sheet, (c) A composite of
what happens at the middle and last sheets.

S.9 Odds and ends
Heinrich Hertz

It is time to set some history straight. In electromagnetism, we have two
fields: electric and magnetic. Our study led us to a conjecture: when changing in time, each field will induce the other field. A description of lecture
demonstrations provided experimental support for that idea. Then James
Clerk Maxwell entered our development. In 1864, Maxwell predicted that
electromagnetic waves exist and travel at the same speed as light. He went
on to suggest that light itself is an electromagnetic wave. We adopted this
view and examined the great spectrum of electromagnetic waves, a spectrum in which (visible) light occupies an interval - but only a small one. This
is a good sequence of events when one is trying to grasp a subject as difficult
as ours, but the historical development was different. Let us examine it.
When Maxwell predicted the existence of electromagnetic waves in 1864,
no one had ever moved electric charges around in a laboratory and
deliberately produced electromagnetic waves. His "prediction" of such
waves was a bonafide prediction, a prophesy, a foretelling. And the identification of light as an electromagnetic wave was likewise a prediction, to be
tested experimentally.
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5.9 ODDS AND ENDS

The world of physics was keenly interested in Maxwell's theory, but
experimental confirmation pre-supposes apparatus capable of performing
the tests. Or it pre-supposes an individual so talented as to be able to devise
new apparatus. In 1864, the technical repertoire of physics could not meet
the challenge of testing Maxwell's predictions. More than twenty years had
to pass. Finally, in the years 1886-8, Heinrich Hertz succeeded.
In those years, Hertz was professor of physics in Karlsruhe, Germany.
His transmitter of electromagnetic waves was fundamentally like ours of
section 5.6: electric charges were sent up and down two vertical wires. For a
detector of such waves, Hertz used the spark which an electric field produces when it tries to push electric charges across a tiny gap in another wire.
The spark was minute, only 1/100 of a millimeter in length, and visible only
to a dark-adapted eye in a perfectly dark room. Even twenty years after
Maxwell's fundamental paper appeared, experimental test was just barely
achievable.
Hertz was able to discern sparks as far as 16 meters from his transmitter.
That was as far as his laboratory space allowed him to go, and it was amply
far enough to show him that an electric field was propagated in a beam-like
fashion, like the straight-line motion of light (in a single uniform
substance).
With a large sheet of the metal zinc (for aluminum was not readily
available in those days), Hertz could reflect the beam from his transmitter.
Moreover, the angles of reflection and incidence were equal. With a huge
prism of solidified asphalt - half a ton of it - he caused the beam to refract.
To explore the "sidedness" of electromagnetic waves, Hertz built the
predecessor of a Polaroid sheet: a human-sized wooden frame across which
were stretched copper wires, forming a grid of parallel lines 3 centimeters
apart. When the grid wires were aligned parallel to the vertical wires of the
transmitter, the beam did not pass through the grid of wires. When the grid
wires were perpendicular to the transmitter wires, the beam passed through
freely. Now we examine the reasons for this behavior.
When the electric field is parallel to the copper grid wires, the field
produces a large electric current. Because copper is such a good conductor,
not much energy is literally absorbed in the wires; rather, most of the
energy is reflected (just as a metal mirror reflects light waves). Thus no
wave passes through the grid of copper wires. When the electric field is
perpendicular to the wires, there is no place to push electric charge, and
hence no reaction on the beam itself occurs: the beam passes through the
grid unaffected.
Hertz's copper grid wires have the same transmitting properties that the
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5 ELECTROMAGNETIC WAVES

Will not
pass

Detector
ignores
this

Will be
passed

Detector
responds to
this component

Before
(behind
wires)

After
(our side
of wires)

Figure 5.19 How Hertz explored the "sidedness" of his electromagnetic waves. The
wooden frame holds a parallel array of copper wires; they are I millimeter thick and
spaced 3 centimeters apart.

iodine-coated polyvinyl alcohol does in a Polaroid sheet. To be sure, this is
an historical inversion: Hertz came first, and our analogy for understanding
Polaroid sheets is based on his insight and experience. Moreover, in the
Polaroid, it is literally absorption (not reflection) that prevents transmission
when the electric field is parallel to the strands.
Hertz even performed an analog of the Polaroid experiment sketched in
figure 5.17. He generated a wave with a vertically-oriented electric field, but
he set his detector to respond only to the horizontal component of an
electric field. Of course, the detector gave no response. Then, between
transmitter and detector, Hertz interposed his grid of wires at 45 degrees.
Voila! The detector responded. Hertz reasoned as sketched in figure 5.19
and as described in the next paragraph.
To understand how the grid of wires affects the electromagnetic wave,
decompose the initial, vertical electric field into components parallel to the
wires and perpendicular to them. The parallel component will be reflected
and hence will not pass through. The perpendicular component will be
allowed to pass. Thus, once past the grid, the electric field is oriented at 45
degrees to the vertical. But that orientation is also 45 degrees with respect
to the horizontal, and so the final electric field has a horizontal component.
Now the detector responds - and establishes that the electromagnetic wave
has a kind of "sidedness."
Hertz observed interference and used it to measure the wavelength of his
electromagnetic waves: 66 centimeters. He had no way to measure the
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5.9 ODDS AND ENDS

frequency of the waves, but he could calculate the frequency of oscillation
of the electric charges in his transmitter; he based the calculation on the size
and structure of the circuitry. Within some generous allowance for error
here, Hertz found the product of frequency and wavelength to be about the
same as the speed of light.
In the conclusion of his 1888 paper, Hertz could proudly write
The experiments described appear to me, at any rate, eminently
adapted to remove any doubt as to the identity of light, radiant
heat, and electromagnetic wave motion. I believe that from now
on we shall have greater confidence in making use of the advantages which this identity enables us to derive both in the study of
optics and of electricity.
Newton and Maxwell

When Heinrich Hertz provided his ingenious experimental support for
Maxwell's theory, he placed the capstone on the edifice of classical physics.
Newton's laws of motion describe the response of a material object to a
force: they describe the way the trajectory bends through space. Maxwell's
laws of electromagnetism describe the mutual interaction of electric and
magnetic fields and the relationship of the fields to their sources: electric
charges at rest or in motion. If one throws in for good measure Newton's
quantitative description of gravitation, one has the laws that constitute the
fundamentals of classical physics. The laws are expressed mathematically as
equations relating the relevant variables, such as velocity or electric field. A
mere handful of such laws provides the theoretical basis for an astounding
technology.
Table 5.1 displays some of the notable events as electromagnetic theory
and its associated technology developed. The telegraph, you will note,
came quite early. In telegraphy, an electric current is sent along a pair of
wires, and no wave ideas are necessary for its development. Just open and
close a switch, much as you can send messages in Morse code by pushing a
doorbell button, thereby closing a circuit and letting electric current ring a
distant bell.
The development of radar began shortly before World War II, but the
war itself gave urgency to the work, in both England and the United States.
The Battle of Britain was won by courage and radar. In the late 1940s,
several physicists used a radar beam to measure the speed of electromagnetic waves. A half-century after Hertz began the project, they
established to high accuracy that what are manifestly electromagnetic waves
travel (in vacuum) at the same speed as visible light (to within the accuracy
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5 ELECTROMAGNETIC WAVES

Table 5.1. Some notable events and dates in the history of electromagnetism
and the technology that evolved from it. The classical physics of Newton
and Maxwell sufficed for the engineering development of all the devices
cited.
1950
Commercial FM radio. Electronic computers.
1940
Commercial television. Radar.
1930
1920
1910
1900

AM radio ("Wireless telegraphy").
1890
Hertz's confirmation of Maxwell's theory.
1880
Telephone. Electric lights. Phonograph.
1870
Maxwell's electromagnetic theory.
1860
1850
1840
Telegraph. Electric generator. Faraday's law.
1830
1820

Electric motor.
Oersted's experiment.

of the experiments, where the uncertainty came only in the sixth figure).
Though neither Maxwell nor Hertz would have had any nagging doubts,
they would have been pleased to see the results.
Electromagnetic theory: a recapitulation

Before we move on to another chapter, we would do well to summarize
what this chapter has taught us about electromagnetic theory. Here are the
essentials.
Physics introduces the electric field E as a conceptual intermediary and as
a physically existing entity. One electric charge produces an electric field in
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5.9 ODDS AND ENDS

the space surrounding it; that field exerts a force on a second electric
charge. In this situation, the electric field is manifestly a conceptual intermediary in the production of a force, but the field also "exists" as an
alteration of the properties of space. The "existence" attribute is most
apparent when we consider radio transmission from earth to a planetary
probe: a propagating electric field exists in otherwise-empty space and
carries the message (even after the transmitter has been shut off).
The operational definition of the electric field is this:
_. ,
r - i j T - . • . r» f° r c e o n charge we place at P
r ,
Value of electric field E at point P = • amount of &chargeFwe use
on q

Physics introduces also the magnetic field. Its direction is defined by the
direction in which a compass needle lines up, but we did not bother to
define its magnitude.
An electric charge - both when at rest and when in motion - produces an
electric field. An electric charge at rest does not produce any magnetic
effect, but an electric charge in motion - an electric current - does produce
a magnetic field.
Once produced by electric charges, electric and magnetic fields can
sustain themselves (in some circumstances, though not invariably). A
changing magnetic field generates an electricfield- that property is codified
as Faraday's law - and a changing electric field generates a magnetic field.
This property of "mutual support" enables electric and magnetic fields to
propagate through space. The moving pattern of fields constitutes an
electromagnetic wave. Such a wave moves through vacuum at the speed of
light.
Indeed, visible light itself is an electromagnetic wave. Moreover, nature
provides us with an entire spectrum of electromagnetic waves. Such waves
range from AM and FM radio at the low frequency, long wavelength end of
the spectrum to X-rays and gamma rays at the high frequency, short
wavelength end. (These are the limits of most practical concern, but waves
beyond them can exist, and some do so.)
The electric field is perpendicular to the wave's propagation direction,
and so is the magnetic field. Thus electromagnetic waves are transverse
waves. More importantly, this perpendicularity gives electromagnetic
waves a "sidedness" and explains the sidedness of light that Polaroid sheets
demonstrate so dramatically.
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5 ELECTROMAGNETIC WAVES

We can see wires and batteries, but the central elements in electricity and
magnetism are the fields, and they are not visible per se. Getting used to
thinking about thefieldsis hard, and it was similarly so for physicists in the
nineteenth century. In his little book Relativity and Its Roots, Banesh Hoffmann put the transformation in viewpoint vividly:
At a time when people were asking what kept bodies moving,
Galileo told them to ask, rather, what brought bodies to rest or
otherwise changed their motion. Faraday initiated a comparable
revolution. At a time when people were concentrating their attention on the visible electromagnetic hardware, he told them to
think, rather, of the rich, invisible content of the surrounding
space - the electromagnetic field.

Additional resources

The charming article, "Hans Christian Oersted - Scientist, Humanist and
Teacher," by J. Rud Nielsen, appears in the reprint volume Physics History from
AAPT Journals, Melba Newell Phillips, editor (American Association of Physics
Teachers, College Park, MD, 1985). The compilation contains also articles on
Faraday and Hertz.
James Clerk Maxwell: A Biography, by Ivan Tolstoy (University of Chicago
Press, Chicago, 1982), ably fulfills the promise of its title.
Joseph Mulligan writes engagingly about "Heinrich Hertz and the Development
of Physics" in an article carrying this title in Physics Today, March 1989, pages
50-7.

Questions
1. (a) Calculate the time interval needed for one full oscillation of the
electric field of yellow light. Take the wavelength of yellow light to be
^yeiiow=5.8xl0~7 meters in vacuum. (You may find it useful to calculate
the frequency as an intermediate step.)
(b) About how many wavelengths are included in the light wave produced by a single atom when it emits a burst of light? The burst occurs over a
time interval of about 0.000000001 seconds (that is, 10"9 seconds).
(c) How long in space is the light wave, that is, the distance from the
front to the end of the entire wave pattern?
Surprising numbers? Yet these numbers are characteristic of the light we
get every sunny day from the sun.
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QUESTIONS

2. Let's return to question 6(6) of chapter 1. Here it is, with one change:
the phrase "particle theory" has been replaced by "wave theory."
A triangular glass prism splits white light into a spectrum, different
colors being bent through different angles. Suggest one or more
ways in which a wave theory of light can explain this phenomenon
and yet be consistent with a single speed of light for all colors in
vacuum.

What is your suggestion now?
3. Suppose we set up a double-slit interference experiment just like the one
we used to measure X for the laser light and then put a separate Polaroid
behind each slit. The light source is a filament lamp (giving white light)
followed by a red filter. The apparatus is sketched in figure 5.20.
Recall that, in a beam of light from a hot filament, the electric field,
though always perpendicular to the propagation direction, does not always
oscillate in the same plane. The orientation swings around, as was
illustrated in figure 5.16.
(a) What will the pattern on the screen be like if both Polaroids are
oriented in the same direction (as though we used a single large
sheet)?
(b) What will the pattern on the screen be like if one Polaroid is aligned
to transmit light with E parallel to the slit and the other Polaroid,
light with E perpendicular to the slit? Why? Can there be destructive interference? (You may find it helpful to draw a sketch showing the electric fields that come from the two slits.)
4. Which "bends" more, red light or violet? Consider two different physical contexts:
Two separate
Polaroids
Lamp

Red filter

Viewing
surface
Figure 5.20
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5 ELECTROMAGNETIC WAVES

(1) Light going through a glass prism.
(2) Light going through a pair of closely-spaced slits.
For each context, respond to the following items.
(a) Which "bends" more, red light or violet?
(b) What is the technical term that should replace the informal notion of
"bends"?
(c) In a few sentences, describe the essence of the theory that we use to
understand the "bending."
5. The speed of red light in a certain transparent material is 2.1xlO8
meters/second. Use the wave theory in the following questions.
(a) For the situation in part (a) of figure 5.21, where a beam of red light
strikes the material, calculate either the semi-chord of the refracted ray or the sine of the angle of refraction. You will need to
make your own drawing of the circle, etc. On your drawing,
indicate the refracted ray and its semi-chord.
White light
Vacuum

Vacuum
Material
Material

Vacuum

(a)

(b)
Figure 5.21

(b) When violet light is incident at the same angle, the semi-chord of the
refracted ray is 0.45/?, where R denotes the radius of the circle.
What is the speed of violet light in the material?
(c) If the material were cut into a prismatic shape, as shown in part (b)
of the figure, would the material produce a normal spectrum (like
that produced by glass)? The alternatives are no spectrum at all or
a reversal of the order of the colors. Explain why you answered as
you did.
6. Figure 5.22 shows two straight copper wires with a flashlight bulb con138
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QUESTIONS

Figure 5.22

nected between them (so that an electron could move along one wire, pass
through the bulb's filament, and then continue along the other wire). This
combination of two wires plus bulb forms a "receiving antenna" for
electromagnetic waves. The transmitting antenna is similar to that shown in
figure 5.9.
If the transmitter (which has vertical antenna arms) produces an electric
field E that oscillates vertically (first pointing straight up, then down, then
up again, and so on), why does the bulb still light up (somewhat, at least)
when the receiving antenna is tipped at 30 degrees, say, relative to the
vertical?
7. Between Alice and the (harmless) laser is a thick sheet of plate glass, as
shown in figure 5.23.
(a) Reproduce the sketch and then draw carefully the continuation of

Laser

Air

Glass

Air

Figure 5.23
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5 ELECTROMAGNETIC WAVES

the light beam as it (or some of it) passes through the glass and
then reaches Alice or fails to reach her. Aim to draw the path
qualitatively faithfully.
(b) Include representative crest lines both in the glass and in the air on
Alice's side. Pay attention to their spacing (at least qualitatively).
For example, should the crest lines have the same spacing in all
three regions? Explain your response.

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