Physics For Scientists And Engineers 6E - part 267

Answers to Quick Quizzes

1065

branch.  The  reactance  of  the  inductor  is  small  so  that
current  exists  in  the  inductor  branch  and  the  lightbulb
glows.  As  the  frequency  increases,  the  inductive  reac-
tance  increases  and  the  capacitive  reactance  decreases.
At high frequencies, more current exists in the capacitor
branch  than  the  inductor  branch  and  the  lightbulb
glows more dimly.

33.7 (a) X

. (b) X

. (c) X

33.8 (c). The cosine of * - is the same as that of $-, so the

cos - factor in Equation 33.31 is the same for both
frequencies. The factor !V

rms

is the same because the

source voltage is fixed. According to Equation 33.27,
changing $ - to * - simply interchanges the values of
X

and X

. Equation 33.25 tells us that such an

interchange does not affect the impedance, so that the
current I

rms

in Equation 33.31 is the same for both

frequencies.

33.9 (c). At resonance, X

. According to Equation

33.25, this gives us Z " R.

33.10(a). The higher the quality factor, the more sensitive the

detector. As you can see from Figure 33.19, when Q "
#

/!# is high, a slight change in the resonance

frequency (as might happen when a small piece of metal
passes through the portal) causes a large change in
current that can be detected easily.

33.11(a) and (e). The current in an inductive circuit

decreases with increasing frequency (see Eq. 33.9). Thus,
an  inductor  connected  in  series  with  a  woofer  blocks
high-frequency  signals  and  passes  low-frequency  signals.
The  current  in  a  capacitive  circuit  increases  with
increasing  frequency  (see  Eq.  33.17).  When  a  capacitor
is connected in series with a tweeter, the capacitor blocks
low-frequency  signals  and  passes  high-frequency  signals.

Chapter 34

Electromagnetic Waves

C H A P T E R O U T L I N E

34.1 Maxwell’s Equations and

Hertz’s Discoveries

34.2 Plane Electromagnetic Waves

34.3 Energy Carried by

Electromagnetic Waves

34.4 Momentum and Radiation

Pressure

34.5 Production of Electromagnetic

Waves by an Antenna

34.6 The Spectrum of

Electromagnetic Waves

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▲

Electromagnetic waves cover a broad spectrum of wavelengths, with waves in various

wavelength ranges having distinct properties. These images of the Crab Nebula show
different structure for observations made with waves of various wavelengths. The images
(clockwise starting from the upper left) were taken with x-rays, visible light, radio waves, and
infrared waves. (upper left—NASA/CXC/SAO; upper right—Palomar Observatory; lower
right—VLA/NRAO; lower left—WM Keck Observatory)

1067

he waves described in Chapters 16, 17, and 18 are mechanical waves. By definition,

the propagation of mechanical disturbances—such as sound waves, water waves, and
waves on a string—requires the presence of a medium. This chapter is concerned with
the properties of electromagnetic waves, which (unlike mechanical waves) can propa-
gate through empty space.

In Section 31.7 we gave a brief description of Maxwell’s equations, which form the

theoretical basis of all electromagnetic phenomena. The consequences of Maxwell’s
equations are far-reaching and dramatic. The Ampère–Maxwell law predicts that a
time-varying electric field produces a magnetic field, just as Faraday’s law tells us that a
time-varying magnetic field produces an electric field.

Astonishingly, Maxwell’s equations also predict the existence of electromagnetic

waves that propagate through space at the speed of light c. This chapter begins with a
discussion of how Heinrich Hertz confirmed Maxwell’s prediction when he generated
and detected electromagnetic waves in 1887. That discovery has led to many practical
communication systems, including radio, television, radar, and opto-electronics. On a
conceptual level, Maxwell unified the subjects of light and electromagnetism by devel-
oping the idea that light is a form of electromagnetic radiation.

Next, we learn how electromagnetic waves are generated by oscillating electric

charges. The waves consist of oscillating electric and magnetic fields at right angles to
each other and to the direction of wave propagation. Thus, electromagnetic waves are
transverse waves. The waves radiated from the oscillating charges can be detected at
great distances. Furthermore, electromagnetic waves carry energy and momentum and
hence can exert pressure on a surface.

The chapter concludes with a look at the wide range of frequencies covered by

electromagnetic waves. For example, radio waves (frequencies of about 10

Hz) are

electromagnetic waves produced by oscillating currents in a radio tower’s transmitting
antenna. Light waves are a high-frequency form of electromagnetic radiation (about
10

Hz) produced by oscillating electrons in atoms.

34.1 Maxwell’s Equations and Hertz’s

Discoveries

In his unified theory of electromagnetism, Maxwell showed that electromagnetic waves
are a natural consequence of the fundamental laws expressed in the following four
equations (see Section 31.7):

(34.1)

(34.2)

B!d

A " 0

E!d

A "

James Clerk

Maxwell

Scottish Theoretical Physicist
(1831–1879)

Maxwell developed the

electromagnetic theory of light

and the kinetic theory of gases,

and explained the nature of

Saturn’s rings and color vision.

Maxwell’s successful

interpretation of the

electromagnetic field resulted in

the field equations that bear his

name. Formidable mathematical

ability combined with great

insight enabled him to lead the

way in the study of

electromagnetism and kinetic

theory. He died of cancer before

he was 50. (North Wind Picture
Archives)

Maxwell’s equations

(34.3)

(34.4)

In the next section we show that Equations 34.3 and 34.4 can be combined to obtain a
wave equation for both the electric field and the magnetic field. In empty space, where
q " 0 and I " 0, the solution to these two equations shows that the speed at which
electromagnetic waves travel equals the measured speed of light. This result led
Maxwell to predict that light waves are a form of electromagnetic radiation.

The experimental apparatus that Hertz used to generate and detect electromag-

netic waves is shown schematically in Figure 34.1. An induction coil is connected to a
transmitter made up of two spherical electrodes separated by a narrow gap. The coil
provides short voltage surges to the electrodes, making one positive and the other neg-
ative. A spark is generated between the spheres when the electric field near either
electrode surpasses the dielectric strength for air (3 $ 10

V/m; see Table 26.1). In a

strong electric field, the acceleration of free electrons provides them with enough
energy to ionize any molecules they strike. This ionization provides more electrons,
which can accelerate and cause further ionizations. As the air in the gap is ionized, it
becomes a much better conductor, and the discharge between the electrodes exhibits
an oscillatory behavior at a very high frequency. From an electric-circuit viewpoint, this
is equivalent to an LC circuit in which the inductance is that of the coil and the capaci-
tance is due to the spherical electrodes.

Because L and C are small in Hertz’s apparatus, the frequency of oscillation is high,

on the order of 100 MHz. (Recall from Eq. 32.22 that

for an LC circuit.)

Electromagnetic waves are radiated at this frequency as a result of the oscillation (and
hence acceleration) of free charges in the transmitter circuit. Hertz was able to detect
these waves using a single loop of wire with its own spark gap (the receiver). Such a
receiver loop, placed several meters from the transmitter, has its own effective induc-
tance, capacitance, and natural frequency of oscillation. In Hertz’s experiment, sparks
were induced across the gap of the receiving electrodes when the frequency of the
receiver was adjusted to match that of the transmitter. Thus, Hertz demonstrated that
the oscillating current induced in the receiver was produced by electromagnetic waves
radiated by the transmitter. His experiment is analogous to the mechanical phenome-
non in which a tuning fork responds to acoustic vibrations from an identical tuning
fork that is oscillating.

% "

√

B!d

s " &

I ' &

(

E!d

s " )

(

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C H A P T E R 3 4 • Electromagnetic Waves

Input

Transmitter

Receiver

Induction

coil

–q

–

Figure 34.1 Schematic diagram of Hertz’s apparatus for

generating and detecting electromagnetic waves. The trans-

mitter consists of two spherical electrodes connected to an

induction coil, which provides short voltage surges to the

spheres, setting up oscillations in the discharge between the

electrodes. The receiver is a nearby loop of wire containing

a second spark gap.

Heinrich Rudolf

Hertz

German Physicist (1857–1894)

Hertz made his most important

discovery of electromagnetic

waves in 1887. After finding that

the speed of an electromagnetic

wave was the same as that of

light, Hertz showed that

electromagnetic waves, like light

waves, could be reflected,

refracted, and diffracted. Hertz

died of blood poisoning at the

age of 36. During his short life,

he made many contributions to

science. The hertz, equal to one

complete vibration or cycle per

second, is named after him.
(Hulton-Deutsch
Collection/CORBIS)

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