SPEED OF EM WAVES IN VACUUM AND IN MATTER

22.5 Speed of EM Waves in Vacuum and in Matter 811
Scanning drum
is rotated through
360 degrees.
X-ray beam
passes through
the body.
X-ray tube
emits x-rays
as the scanner
rotates around
the body.
X-ray detector
records intensity
of x-rays
transmitted
through the body.
Movable bed
allows any part
of the body
to be scanned.
Figure 22.12 Apparatus used for a CAT scan.
Gamma rays were first observed in the decay of radioactive nuclei on Earth.
Pulsars, neutron stars, black holes, and explosions of supernovae are sources of gamma
rays that travel toward Earth, but—fortunately for us—gamma rays are absorbed by the
atmosphere. Only when detectors were placed high in the atmosphere and above it by
using balloons and satellites did the science of gamma-ray astronomy develop. In the
late 1960s, scientists first observed bursts of gamma rays from deep space that last for
times ranging from a fraction of a second to a few minutes; these bursts occur about
once a day. A gamma-ray burst can emit more energy in 10 s than the Sun will emit in
its entire lifetime. The source of the gamma-ray bursts is still under investigation.
22.5 SPEED OF EM WAVES IN VACUUM
AND IN MATTER
Light travels so fast that it is not obvious that it takes any time at all to go from one place
to another. Since high-precision electronic instruments were not available, early measurements
of the speed of light had to be cleverly designed. In 1849, French scientist
Armand Hippolyte Louis Fizeau (1819–1896) measured the speed of visible light to be
approximately 3 × 10 8 m/s. Fizeau’s experiment used a notched wheel (Fig. 22.13).
When the apparatus is correctly aligned, a beam of light passes through one of the
notches in the wheel, travels a long distance (over 8 km) to a mirror, reflects, and passes
back through the same notch to the observer. When the wheel is made to rotate, the notch

812 Chapter 22 Electromagnetic Waves
8.6 km
Semitransparent
mirror
Mirror
Figure 22.13 The apparatus
used by Fizeau in 1849 to measure
the speed of light.
Observer
Beam
of light
w
Rotating
notched
wheel
moves out of position and the reflected beam is interrupted by the wheel. As the angular
velocity of the wheel is increased, it reaches a value w where the next notch moves into
position just in time to allow the reflected beam to pass through. The observer can see the
reflected beam for an integral multiple of w, since any of the equally spaced notches
allow the reflected beam to pass through. The speed of light can be determined from a
measurement of the angular velocities at which the observer sees the reflected beam.
In Chapters 11 and 12 we saw that the speed of a mechanical wave depends on
properties of the wave medium. Sound travels faster through steel than it does through
water and faster through water than through air. In every case, the wave speed depended
on two characteristics of the wave medium: one that characterizes the restoring force
and another that characterizes the inertia.
Unlike mechanical waves, electromagnetic waves can travel through vacuum; they
do not require a material medium. Light reaches Earth from galaxies billions of lightyears
away, traveling the vast distances between galaxies without problem; but a sound
wave can’t even travel a few meters between two astronauts on a space walk, since there
is no air or other medium to sustain a sound wave’s pressure variations. What, then,
determines the speed of light in vacuum?
Looking back at the laws that describe electric and magnetic fields, we find two
universal constants. One of them is the permittivity of free space 0 , found in
Coulomb’s law and Gauss’s law; it is associated with the electric field. The second is the
permeability of free space m 0 , found in Ampère’s law; it is associated with the magnetic
field. Since these are the only two quantities that can determine the speed of light in
vacuum, there must be a combination of them that has the dimensions of speed.
The values of these constants in SI units are
0 = 8.85 × 10 –12 2
C
N •m 2
and
m 0 = 4p × 10 –7 T •m
A
The tesla can be written in terms of other SI units. Using F = qv × B as a guide,
N
1 T = 1
C• m/s
Multiplying 0 × m 0 gives
0 m 0 = 8.85 × 10 –12 2
—C
–N
2 × 4p × 10 –7 –N •–m
•m
—C•(–m /s)•(—C/s)
= 1.11 × 10 –17 s
2
m
2
To end up with m/s, we need to take the reciprocal of the square root:
1
= 3.00 × 108 m/s
0 m 0

22.5 Speed of EM Waves in Vacuum and in Matter 813
The dimensional analysis done here leaves the possibility of a multiplying factor
such as 1 2 or p. In the mid-nineteenth century, Scottish physicist James Clerk Maxwell
proved mathematically that an electromagnetic wave—a wave consisting of oscillating
electric and magnetic fields propagating through space—could exist in vacuum.
Starting from Maxwell’s equations (Section 22.2), he derived the wave equation, an
equation of a special mathematical form that describes wave propagation for any kind
of wave. In the place of the wave speed appeared ( 0 m 0 ) –1/2 . Using the values of 0 and
m 0 that had been measured in 1856, Maxwell showed that electromagnetic waves in vacuum
travel at 3.00 × 10 8 m/s—very close to what Fizeau measured. Maxwell’s derivation
was the first evidence that light is an electromagnetic wave.
The speed of electromagnetic waves in vacuum is represented by the symbol c (for
the Latin celeritas, “speed”).
Speed of electromagnetic waves in vacuum:
1
c = 0m = 3.00 × 10 8 m/s (22-3)
0
While c is usually called the speed of light, it is the speed of any electromagnetic wave
in vacuum, regardless of frequency or wavelength, not just the speed for frequencies
visible to humans.
Example 22.2
Light Travel Time from a “Nearby” Supernova
A supernova is an exploding star; a supernova is Solution The time for light to travel a distance d at
billions of times brighter than an ordinary star. speed c is
Most supernovae occur in distant galaxies and cannot be
observed with the naked eye. The last two supernovae visible
to the naked eye occurred in 1604 and 1987.
∆t = d 21
c =
1.6
× 10
m
= 5.33 × 10 12 s
3
8
.00
× 10
m/s
Supernova SN1987a (Fig. 22.14) occurred 1.6 × 10 21 To get a better idea how long that is, we convert seconds
m
to years:
from Earth. When did the explosion occur?
5.33 × 10 12 1 yr
s × = 170,000 yr
Strategy The light from the supernova travels at speed c.
3.156 × 10 7 s
The time that it takes light to travel a distance 1.6 × 10 21 m
tells us how long ago the explosion occurred.
Discussion When we look at the stars, the light we
see was radiated by the stars long ago. By looking at distant
galaxies, astronomers get a glimpse of the universe
in the past. Beyond the Sun, the closest star to Earth is
about 4 ly (light-years) away, which means that it takes
light 4 yr to reach us from that star. The most distant
galaxies observed are at a distance of over 10 10 ly; looking
at them, we see over 10 billion yr into the past.
Figure 22.14
Photo of the sky after light from Supernova SN1987a reached
Earth.
Practice Problem 22.2
A Light-Year
A light-year is the distance traveled by light (in vacuum)
in one Earth year. Find the conversion factor from lightyears
to meters.

814 Chapter 22 Electromagnetic Waves
The speed of an EM wave
through matter is less than c.
Speed of Light in Matter
When an EM wave travels through a material medium, it travels at a speed v that is less
than c. For example, visible light travels through glass at speeds between about
1.6 × 10 8 m/s and 2.0 × 10 8 m/s, depending on the type of glass and the frequency of
the light. Instead of specifying the speed, it is common to specify the index of refraction
n:
Index of refraction:
n = v
c (22-4)
A wave passing from one
medium into another changes
wavelength but retains the
same frequency.
Refraction refers to the bending of a wave as it passes from one medium to another;
we study refraction in detail in Section 23.3. Since the index of refraction is a ratio of two
speeds, it is a dimensionless number. For glass in which light travels at 2.0 × 10 8 m/s, the
index of refraction is
n = 3 8
. 0 × 10
m/
s
= 1.5
8
2.
0 × 10
m/
s
The speed of light in air (at 1 atm) is only slightly less than c; the index of refraction
of air is 1.0003. Most of the time this 0.03% difference is not important, so we can
use c as the speed of light in air. The speed of visible light in an optically transparent
medium is less than c, so the index of refraction is greater than 1.
When an EM wave passes from one medium to another, the frequency and wavelength
cannot both remain unchanged since
v = fl
As is the case with mechanical waves, it is the wavelength that changes; the frequency
remains the same. The incoming wave (with frequency f ) causes charges in the atoms at
the boundary to oscillate with the same frequency f, just as for the charges in an
antenna. The oscillating charges at the boundary radiate an EM wave at that same frequency
into the second medium. Therefore, the electric and magnetic fields in the second
medium must oscillate at the same frequency as the fields in the first medium. In
just the same way, if a transverse wave of frequency f traveling down a string reaches a
point at which an abrupt change in wave speed occurs, the incident wave makes that
point oscillate up and down at the same frequency f as any other point on the string. The
oscillation of that point sends a wave of the same frequency to the other side of the
string. Since the wave speed has changed but the frequency is the same, the wavelength
has changed as well.
We sometimes need to find the wavelength l of an EM wave in a medium of index
n, given its wavelength l 0 in vacuum. Since the frequencies are equal,
c v
f = = l l
Solving for l gives
0
l = v c l 0 = l n 0 (22-5)
Since n > 1, the wavelength is shorter than the wavelength in vacuum. The wave travels
more slowly in the medium than in vacuum; since the wavelength is the distance traveled
by the wave in one period T = 1/f, the wavelength in the medium is shorter.
If blue light of wavelength l 0 = 480 nm enters glass that has an index of refraction
of 1.5, it is still visible light, even though its wavelength in glass is 320 nm; it has not
been transformed into UV radiation. When the light enters the eye, it has the same frequency
and wavelength in the fluid in the eye regardless of how many material media it
has passed through, since the frequency remains the same at each boundary.

22.6 Characteristics of Electromagnetic Waves in Vacuum 815
Example 22.3
Wavelength Change from Glass to Water
The index of refraction of glass is 1.50 and that of water
is 1.33. If light of wavelength 285 nm in glass passes into
water, what is the wavelength in the water?
Strategy The key is to remember that the frequency is
the same as the wave passes from one medium to another.
Solution Frequency, wavelength, and speed are related
by
v = lf
Solving for frequency, f = v/l. Since the frequencies are
equal,
v l w w = v l g
g
The speed of light in a material is v = c/n. Solving for l w
and substituting v = c/n gives
lg c
l w = v w v
=
g n
× n gl g
w c
= 1.50 × 285nm
= 321 nm
1. 33
Discussion Water has a smaller index of refraction, so
the speed of light in water is greater than in glass. Since
wavelength is the distance traveled in one period, the
wavelength in water is longer (321 nm > 285 nm).
Practice Problem 22.3
from Air to Water
Wavelength Change
The speed of visible light in water is 2.25 × 10 8 m/s.
When light of wavelength 592 nm in air passes into
water, what is its wavelength in water?
Dispersion
Although EM waves of every frequency travel through vacuum at the same speed c, the
speed of EM waves in a material medium does depend on frequency. Therefore, the
index of refraction is not a constant for a given material; it is a function of frequency.
Variation of the speed of a wave with frequency is called dispersion. Dispersion causes
white light to separate into colors when it passes through a glass prism (Fig. 22.15). The
dispersion of the light into different colors arises because each color travels at a slightly
different speed in the same medium.
A nondispersive medium is one for which the variation in the index of refraction is
negligibly small for the range of frequencies of interest. No medium (apart from vacuum)
is truly nondispersive, but many can be treated as nondispersive for a restricted
range of frequencies. For most optically transparent materials, the index of refraction
increases with increasing frequency; blue light travels more slowly through glass than
does red light. In other parts of the EM spectrum, or even for visible light in unusual
materials, n can decrease with increasing frequency instead.
Figure 22.15 A prism separates
a beam of white light (coming
in from the left) into the
colors of the spectrum.
22.6 CHARACTERISTICS OF ELECTROMAGNETIC
WAVES IN VACUUM
The various characteristics of EM waves in vacuum can be derived from the basic laws
of electromagnetism (Maxwell’s equations, Section 22.2). Such a derivation requires
higher level mathematics, so we state the characteristics without proof.
• EM waves in vacuum travel at speed c = 3.00 × 10 8 m/s, independent of frequency.
The speed is also independent of amplitude.
• The electric and magnetic fields oscillate at the same frequency. Thus, a single frequency
f and a single wavelength l = c/f pertain to both the electric and magnetic
fields of the wave.
• The electric and magnetic fields oscillate in phase with each other. That is, at a
given instant, the electric and magnetic fields are at their maximum magnitudes at a