chapter 38 Diffraction patterns and polarization
slits, separated by 1.20 mm, and falls on a sheet of pho-
tographic film 1.40 m away. The exposure time is cho-
sen so that the film stays unexposed everywhere except
at the central region of each bright fringe. (a) Find the
distance between these interference maxima. The film
is printed as a transparency; it is opaque everywhere
except at the exposed lines. Next, the same beam of
laser light is directed through the transparency and
allowed to fall on a screen 1.40 m beyond. (b) Argue
that several narrow, parallel, bright regions, separated
by 1.20 mm, appear on the screen as real images of the
original slits. (A similar train of thought, at a soccer
game, led Dennis Gabor to invent holography.)
37. A beam of bright red light of wavelength 654 nm passes
through a diffraction grating. Enclosing the space
beyond the grating is a large semicylindrical screen cen-
tered on the grating, with its axis parallel to the slits in
the grating. Fifteen bright spots appear on the screen.
Find (a) the maximum and (b) the minimum possible
values for the slit separation in the diffraction grating.
Section 38.5 Diffraction of X-Rays by Crystals
38. If the spacing between planes of atoms in a NaCl crys-
tal is 0.281 nm, what is the predicted angle at which
0.140-nm x-rays are diffracted in a first-order maximum?
39. Potassium iodide (KI) has the same crystalline struc-
ture as NaCl, with atomic planes separated by 0.353 nm.
A monochromatic x-ray beam shows a first-order dif-
fraction maximum when the grazing angle is 7.60°.
Calculate the x-ray wavelength.
40. Monochromatic x-rays (l 5 0.166 nm) from a nickel
target are incident on a potassium chloride (KCl) crys-
tal surface. The spacing between planes of atoms in
KCl is 0.314 nm. At what angle (relative to the surface)
should the beam be directed for a second-order maxi-
mum to be observed?
41. The first-order diffraction maximum is observed at
12.6° for a crystal having a spacing between planes of
atoms of 0.250 nm. (a) What wavelength x-ray is used to
observe this first-order pattern? (b) How many orders
can be observed for this crystal at this wavelength?
Section 38.6 Polarization of light Waves
Problem 62 in Chapter 34 can be assigned with this
42. Why is the following situation impossible? A technician is
measuring the index of refraction of a solid material
by observing the polarization of light reflected from
its surface. She notices that when a light beam is pro-
jected from air onto the material surface, the reflected
light is totally polarized parallel to the surface when
the incident angle is 41.0°.
43. Plane-polarized light is incident on a single polarizing
disk with the direction of E
parallel to the direction
of the transmission axis. Through what angle should
the disk be rotated so that the intensity in the transmit-
ted beam is reduced by a factor of (a) 3.00, (b) 5.00,
and (c) 10.0?
nonpitted surfaces, they are detected with constant high
intensity. If the main beam wanders off the track, how-
ever, one of the side beams begins to strike pits on the
information track and the reflected light diminishes.
This change is used with an electronic circuit to guide
the beam back to the desired location. Assume the laser
light has a wavelength of 780 nm and the diffraction
grating is positioned 6.90 mm from the disk. Assume
the first-order beams are to fall on the CD 0.400 mm on
either side of the information track. What should be the
number of grooves per millimeter in the grating?
30. A grating with 250 grooves/mm is used with an incan-
descent light source. Assume the visible spectrum to
range in wavelength from 400 nm to 700 nm. In how
many orders can one see (a) the entire visible spec-
trum and (b) the short-wavelength region of the visible
31. A diffraction grating has 4 200 rulings/cm. On a screen
2.00 m from the grating, it is found that for a particu-
lar order m, the maxima corresponding to two closely
spaced wavelengths of sodium (589.0 nm and 589.6 nm)
are separated by 1.54 mm. Determine the value of m.
32. The hydrogen spectrum includes a red line at 656nm
and a blue-violet line at 434 nm. What are the angu-
lar separations between these two spectral lines for
all visible orders obtained with a diffraction grating
that has 4 500 grooves/cm?
33. Light from an argon laser strikes a diffraction grating
that has 5 310 grooves per centimeter. The central and
first-order principal maxima are separated by 0.488 m
on a wall 1.72 m from the grating. Determine the wave-
length of the laser light.
34. Show that whenever white light is passed through a dif-
fraction grating of any spacing size, the violet end of
the spectrum in the third order on a screen always over-
laps the red end of the spectrum in the second order.
35. Light of wavelength 500 nm is incident normally on
a diffraction grating. If the third-order maximum of
the diffraction pattern is observed at 32.0°, (a) what is
the number of rulings per centimeter for the grating?
(b) Determine the total number of primary maxima
that can be observed in this situation.
36. A wide beam of laser light with a wavelength of
632.8nm is directed through several narrow parallel
(arbitrary). Calculate the transmitted intensity I
5 20.0°, u
5 40.0°, and u
5 60.0°. Hint: Make
repeated use of Malus’s law.
52. Two polarizing sheets are placed together with their
transmission axes crossed so that no light is trans-
mitted. A third sheet is inserted between them with
its transmission axis at an angle of 45.0° with respect
to each of the other axes. Find the fraction of inci-
dent unpolarized light intensity transmitted by the
three-sheet combination. (Assume each polarizing
sheet is ideal.)
53. In a single-slit diffraction pattern, assuming each side
maximum is halfway between the adjacent minima,
find the ratio of the intensity of (a) the first-order side
maximum and (b) the second-order side maximum to
the intensity of the central maximum.
54. Laser light with a wavelength of 632.8 nm is directed
through one slit or two slits and allowed to fall on a
screen 2.60 m beyond. Figure P38.54 shows the pat-
tern on the screen, with a centimeter ruler below it.
(a) Did the light pass through one slit or two slits?
Explain how you can determine the answer. (b) If one
slit, find its width. If two slits, find the distance between
10 11 12 13
55. In water of uniform depth, a wide pier is supported on
pilings in several parallel rows 2.80 m apart. Ocean
waves of uniform wavelength roll in, moving in a direc-
tion that makes an angle of 80.0° with the rows of pil-
ings. Find the three longest wavelengths of waves that
are strongly reflected by the pilings.
56. The second-order dark fringe in a single-slit diffrac-
tion pattern is 1.40 mm from the center of the central
maximum. Assuming the screen is 85.0 cm from a slit
of width 0.800 mm and assuming monochromatic
incident light, calculate the wavelength of the incident
57. Light from a helium–neon laser (l 5 632.8 nm) is inci-
dent on a single slit. What is the maximum width of the
slit for which no diffraction minima are observed?
58. Two motorcycles separated laterally by 2.30 m are
approaching an observer wearing night-vision gog-
gles sensitive to infrared light of wavelength 885 nm.
(a) Assume the light propagates through perfectly
steady and uniform air. What aperture diameter
is required if the motorcycles’ headlights are to be
resolved at a distance of 12.0km? (b)Comment on
how realistic the assumption in part (a)is.
44. The angle of incidence of a light beam onto a reflect-
ing surface is continuously variable. The reflected ray
in air is completely polarized when the angle of inci-
dence is 48.0°. What is the index of refraction of the
45. Unpolarized light passes through two ideal Polaroid
sheets. The axis of the first is vertical, and the axis of
the second is at 30.0° to the vertical. What fraction of
the incident light is transmitted?
46. Two handheld radio transceivers with dipole antennas
are separated by a large fixed distance. If the transmit-
ting antenna is vertical, what fraction of the maximum
received power will appear in the receiving antenna
when it is inclined from the vertical by (a) 15.0°,
(b) 45.0°, and (c) 90.0°?
47. You use a sequence of ideal polarizing filters, each
with its axis making the same angle with the axis of
the previous filter, to rotate the plane of polarization
of a polarized light beam by a total of 45.0°. You wish
to have an intensity reduction no larger than 10.0%.
(a) How many polarizers do you need to achieve
your goal? (b) What is the angle between adjacent
48. An unpolarized beam of light is incident on a stack of
ideal polarizing filters. The axis of the first filter is per-
pendicular to the axis of the last filter in the stack. Find
the fraction by which the transmitted beam’s intensity
is reduced in the three following cases. (a) Three filters
are in the stack, each with its transmission axis at 45.0°
relative to the preceding filter. (b) Four filters are in
the stack, each with its transmission axis at 30.0° rela-
tive to the preceding filter. (c) Seven filters are in the
stack, each with its transmission axis at 15.0° relative to
the preceding filter. (d) Comment on comparing the
answers to parts (a), (b), and (c).
49. The critical angle for total internal reflection for sap-
phire surrounded by air is 34.4°. Calculate the polar-
izing angle for sapphire.
50. For a particular transparent medium surrounded by
air, find the polarizing angle u
in terms of the critical
angle for total internal reflection u
51. Three polarizing plates whose planes are parallel are
centered on a common axis. The directions of the
transmission axes relative to the common vertical
direction are shown in Figure P38.51. A linearly polar-
ized beam of light with plane of polarization parallel
to the vertical reference direction is incident from
the left onto the first disk with intensity I
5 10.0 units
chapter 38 Diffraction patterns and polarization
polarized when it is at 36.0° with respect to the surface,
what is the wavelength of the refracted ray?
64. Iridescent peacock feathers are shown in Figure
P38.64a. The surface of one microscopic barbule is
composed of transparent keratin that supports rods
of dark brown melanin in a regular lattice, repre-
sented in Figure P38.64b. (Your fingernails are made
of keratin, and melanin is the dark pigment giving
color to human skin.) In a portion of the feather that
can appear turquoise (blue-green), assume the mela-
nin rods are uniformly separated by 0.25 mm, with
air between them. (a) Explain how this structure can
appear turquoise when it contains no blue or green
pigment. (b) Explain how it can also appear violet if
light falls on it in a different direction. (c) Explain
how it can present different colors to your two eyes
simultaneously, which is a characteristic of iridescence.
(d) A compact disc can appear to be any color of the
rainbow. Explain why the portion of the feather in
Figure P38.64b cannot appear yellow or red. (e) What
could be different about the array of melanin rods in a
portion of the feather that does appear to be red?
60. Two wavelengths l and l 1 Dl (with Dl ,, l) are inci-
dent on a diffraction grating. Show that the angular
separation between the spectral lines in the mth-order
where d is the slit spacing and m is the order number.
61. Review. A beam of 541-nm light is incident on a dif-
fraction grating that has 400 grooves/mm. (a) Deter-
mine the angle of the second-order ray. (b) What If? If
the entire apparatus is immersed in water, what is the
new second-order angle of diffraction? (c) Show that
the two diffracted rays of parts (a) and (b) are related
through the law of refraction.
62. Why is the following situation impossible? A technician is
sending laser light of wavelength 632.8 nm through a
pair of slits separated by 30.0 mm. Each slit is of width
2.00 mm. The screen on which he projects the pattern
is not wide enough, so light from the m 5 15 inter-
ference maximum misses the edge of the screen and
passes into the next lab station, startling a coworker.
63. A 750-nm light beam in air hits the flat surface of a
certain liquid, and the beam is split into a reflected ray
and a refracted ray. If the reflected ray is completely
Problems 65 and 66.
graph of intensity versus f has a horizontal tangent at
maxima and also at minima.
72. How much diffraction spreading does a light beam
undergo? One quantitative answer is the full width at
half maximum of the central maximum of the single-slit
Fraunhofer diffraction pattern. You can evaluate this
angle of spreading in this problem. (a) In Equation
38.2, define f5 pa sin u/l and show that at the point
where I 5 0.5I
we must have f5!2
sin f. (b) Let
5 sin f and y
. Plot y
on the same
set of axes over a range from f 5 1 rad to f 5 p/2 rad.
Determine f from the point of intersection of the
two curves. (c) Then show that if the fraction l/a is
not large, the angular full width at half maximum of
the central diffraction maximum is u 5 0.885l/a.
(d) What If? Another method to solve the transcen-
dental equation f5 !2
sin f in part (a) is to guess
a first value of f, use a computer or calculator to see
how nearly it fits, and continue to update your estimate
until the equation balances. How many steps (itera-
tions) does this process take?
73. Two closely spaced wavelengths of light are incident on a
diffraction grating. (a) Starting with Equation 38.7, show
that the angular dispersion of the grating is given by
d cos u
(b) A square grating 2.00 cm on each side containing
8 000 equally spaced slits is used to analyze the spec-
trum of mercury. Two closely spaced lines emitted
by this element have wavelengths of 579.065 nm and
576.959 nm. What is the angular separation of these
two wavelengths in the second-order spectrum?
74. Light of wavelength 632.8 nm illuminates a single slit,
and a diffraction pattern is formed on a screen 1.00 m
from the slit. (a) Using the data in the following table,
plot relative intensity versus position. Choose an appro-
priate value for the slit width a and, on the same graph
used for the experimental data, plot the theoretical
expression for the relative intensity
where f 5 (pa sin u)/l. (b) What value of a gives the
best fit of theory and experiment?
Position Relative to
Central Maximum (mm)
your eyes are 5.00mm in diameter, and the average
wavelength of the light coming from the screen is
550 nm. Calculate the ratio of the minimum viewing
distance to the vertical dimension of the picture such
that you will not be able to resolve the lines.
68. A pinhole camera has a small circular aperture of
diameter D. Light from distant objects passes through
the aperture into an otherwise dark box, falling on a
screen located a distance L away. If D is too large, the
display on the screen will be fuzzy because a bright
point in the field of view will send light onto a circle of
diameter slightly larger than D. On the other hand, if
D is too small, diffraction will blur the display on the
screen. The screen shows a reasonably sharp image
if the diameter of the central disk of the diffraction
pattern, specified by Equation 38.6, is equal to D at
the screen. (a) Show that for monochromatic light
with plane wave fronts and L .. D, the condition for
a sharp view is fulfilled if D2 5 2.44lL. (b) Find the
optimum pinhole diameter for 500-nm light projected
onto a screen 15.0 cm away.
69. The scale of a map is a number of kilometers per centi-
meter specifying the distance on the ground that any
distance on the map represents. The scale of a spectrum
is its dispersion, a number of nanometers per centime-
ter, specifying the change in wavelength that a distance
across the spectrum represents. You must know the
dispersion if you want to compare one spectrum with
another or make a measurement of, for example, a Dop-
pler shift. Let y represent the position relative to the
center of a diffraction pattern projected onto a flat
screen at distance L by a diffraction grating with slit
spacing d. The dispersion is dl/dy. (a) Prove that the
dispersion is given by
(b) A light with a mean wavelength of 550 nm is ana-
lyzed with a grating having 8 000 rulings/cm and pro-
jected onto a screen 2.40 m away. Calculate the disper-
sion in first order.
70. (a) Light traveling in a medium of index of refraction
is incident at an angle u on the surface of a medium
of index n
. The angle between reflected and refracted
rays is b. Show that
(b) What If? Show that this expression for tan u reduces
to Brewster’s law when b 5 90°.
71. The intensity of light in a diffraction pattern of a single
slit is described by the equation
where f 5 (pa sin u)/l. The central maximum is at
f 5 0, and the side maxima are approximately at f 5
2p for m 5 1, 2, 3, . . . . Determine more pre-
cisely (a) the location of the first side maximum,
where m 5 1, and (b) the location of the second side
maximum. Suggestion: Observe in Figure 38.6a that the
chapter 38 Diffraction patterns and polarization
sources, separated laterally by 20.0 cm, are behind the
slit. What must be the maximum distance between the
plane of the sources and the slit if the diffraction pat-
terns are to be resolved? In this case, the approxima-
tion sin u < tan u is not valid because of the relatively
small value of a/l.
78. In Figure P38.78, suppose the transmission axes of
the left and right polarizing disks are perpendicular
to each other. Also, let the center disk be rotated on
the common axis with an angular speed v. Show that
if unpolarized light is incident on the left disk with an
, the intensity of the beam emerging from
the right disk is
This result means that the intensity of the emerging
beam is modulated at a rate four times the rate of
rotation of the center disk. Suggestion: Use the trigo-
nometric identities cos2 u5
and sin2 u5
u = vt
79. Consider a light wave passing through a slit and propa-
gating toward a distant screen. Figure P38.79 shows the
intensity variation for the pattern on the screen. Give a
mathematical argument that more than 90% of the
transmitted energy is in the central maximum of the
diffraction pattern. Suggestion: You are not expected to
calculate the precise percentage, but explain the steps
of your reasoning. You may use the identification
75. Figure P38.75a is a three-dimensional sketch of a bire-
fringent crystal. The dotted lines illustrate how a thin,
parallel-faced slab of material could be cut from the
larger specimen with the crystal’s optic axis parallel to
the faces of the plate. A section cut from the crystal
in this manner is known as a retardation plate. When a
beam of light is incident on the plate perpendicular
to the direction of the optic axis as shown in Figure
P38.75b, the O ray and the E ray travel along a single
straight line, but with different speeds. The figure
shows the wave fronts for the two rays. (a) Let the thick-
ness of the plate be d. Show that the phase difference
between the O ray and the E ray after traveling the
thickness of the plate is
where l is the wavelength in air. (b) In a particular
case, the incident light has a wavelength of 550 nm.
Find the minimum value of d for a quartz plate for
which u 5 p/2. Such a plate is called a quarter-wave
plate. Use values of n
from Table 38.1.
76. A spy satellite can consist of a large-diameter concave
mirror forming an image on a digital-camera detec-
tor and sending the picture to a ground receiver by
radio waves. In effect, it is an astronomical telescope
in orbit, looking down instead of up. (a) Can a spy sat-
ellite read a license plate? (b) Can it read the date on
a dime? Argue for your answers by making an order-
of-magnitude calculation, specifying the data you
77. Suppose the single slit in Figure 38.4 is 6.00 cm wide
and in front of a microwave source operating at
7.50 GHz. (a) Calculate the angle for the first mini-
mum in the diffraction pattern. (b) What is the rela-
tive intensity I/I
at u 5 15.0°? (c) Assume two such
The Compact Muon Solenoid
(CMS) Detector is part of
the Large Hadron Collider
at the European Laboratory
for Particle Physics operated
by CERN. It is one of several
detectors that search for
elementary particles. For a
sense of scale, the green
structure to the left of the
detector and extending to the
top is five stories high.
At the end of the 19th century, many scientists believed they had learned most of
what there was to know about physics. Newton’s laws of motion and theory of universal
gravitation, Maxwell’s theoretical work in unifying electricity and magnetism, the laws of ther-
modynamics and kinetic theory, and the principles of optics were highly successful in explaining a
variety of phenomena.
At the turn of the 20th century, however, a major revolution shook the world of physics. In 1900,
Max Planck provided the basic ideas that led to the formulation of the quantum theory, and in 1905,
Albert Einstein formulated his special theory of relativity. The excitement of the times is captured in
Einstein’s own words: “It was a marvelous time to be alive.” Both theories were to have a profound
effect on our understanding of nature. Within a few decades, they inspired new developments in the
fields of atomic physics, nuclear physics, and condensed-matter physics.
In Chapter 39, we shall introduce the special theory of relativity. The theory provides us with a
new and deeper view of physical laws. Although the predictions of this theory often violate our com-
mon sense, the theory correctly describes the results of experiments involving speeds near the speed
of light. The extended version of this textbook, Physics for Scientists and Engineers with Modern Phys-
ics, covers the basic concepts of quantum mechanics and their application to atomic and molecular
physics. In addition, we introduce condensed matter physics, nuclear physics, particle physics, and
cosmology in the extended version.
Even though the physics that was developed during the 20th century has led to a multitude of
important technological achievements, the story is still incomplete. Discoveries will continue to
evolve during our lifetimes, and many of these discoveries will deepen or refine our understand-
ing of nature and the Universe around us. It is still a “marvelous time to be alive.”
P A A R R T
Our everyday experiences and observations involve objects that move at speeds much
less than the speed of light. Newtonian mechanics was formulated by observing and
describing the motion of such objects, and this formalism is very successful in describing a
wide range of phenomena that occur at low speeds. Nonetheless, it fails to describe properly
the motion of objects whose speeds approach that of light.
Experimentally, the predictions of Newtonian theory can be tested at high speeds by
accelerating electrons or other charged particles through a large electric potential dif-
ference. For example, it is possible to accelerate an electron to a speed of 0.99c (where c
is the speed of light) by using a potential difference of several million volts. According to
Newtonian mechanics, if the potential difference is increased by a factor of 4, the electron’s
kinetic energy is four times greater and its speed should double to 1.98c. Experiments show,
however, that the speed of the electron—as well as the speed of any other object in the Uni-
verse—always remains less than the speed of light, regardless of the size of the accelerating
voltage. Because it places no upper limit on speed, Newtonian mechanics is contrary to
modern experimental results and is clearly a limited theory.
In 1905, at the age of only 26, Einstein published his special theory of relativity. Regard-
ing the theory, Einstein wrote:
39.1 The Principle of Galilean
39.2 The Michelson–Morley
39.3 Einstein’s Principle of
39.4 Consequences of the
Special Theory of Relativity
39.5 The Lorentz Transformation
39.6 The Lorentz Velocity
39.7 Relativistic Linear
39.8 Relativistic Energy
39.9 The General Theory of
c h a p p t t e r
Standing on the shoulders of a giant.
David Serway, son of one of the
authors, watches over two of his
children, Nathan and Kaitlyn, as they
frolic in the arms of Albert Einstein’s
statue at the Einstein memorial in
Washington, D.C. It is well known
that Einstein, the principal architect
of relativity, was very fond of
39.1 the principle of Galilean relativity
The relativity theory arose from necessity, from serious and deep contradictions in the
old theory from which there seemed no escape. The strength of the new theory lies in
the consistency and simplicity with which it solves all these difficulties.1
Although Einstein made many other important contributions to science, the special the-
ory of relativity alone represents one of the greatest intellectual achievements of all time.
With this theory, experimental observations can be correctly predicted over the range of
speeds from v 5 0 to speeds approaching the speed of light. At low speeds, Einstein’s theory
reduces to Newtonian mechanics as a limiting situation. It is important to recognize that
Einstein was working on electromagnetism when he developed the special theory of relativ-
ity. He was convinced that Maxwell’s equations were correct, and to reconcile them with
one of his postulates, he was forced into the revolutionary notion of assuming that space
and time are not absolute.
This chapter gives an introduction to the special theory of relativity, with emphasis on
some of its predictions. In addition to its well-known and essential role in theoretical phys-
ics, the special theory of relativity has practical applications, including the design of nuclear
power plants and modern global positioning system (GPS) units. These devices depend on
relativistic principles for proper design and operation.
39.1 The Principle of Galilean Relativity
To describe a physical event, we must establish a frame of reference. You should
recall from Chapter 5 that an inertial frame of reference is one in which an object is
observed to have no acceleration when no forces act on it. Furthermore, any frame
moving with constant velocity with respect to an inertial frame must also be an
There is no absolute inertial reference frame. Therefore, the results of an exper-
iment performed in a vehicle moving with uniform velocity must be identical to the
results of the same experiment performed in a stationary vehicle. The formal state-
ment of this result is called the principle of Galilean relativity:
The laws of mechanics must be the same in all inertial frames of reference.
Let’s consider an observation that illustrates the equivalence of the laws of mechan-
ics in different inertial frames. The pickup truck in Figure 39.1a moves with a
WWPrinciple of Galilean relativity
1A. Einstein and L. Infield, The Evolution of Physics (New York: Simon and Schuster, 1961).
watch the path of a thrown ball
and obtain different results.
The observer in the moving truck
sees the ball travel in a vertical
path when thrown upward.
The Earth-based observer sees
the ball’s path as a parabola.
An event occurs at
a point P. The event is seen by two
observers in inertial frames S and
S9, where S9 moves with a velocity
relative to S.
Suppose some physical phenomenon, which we call an event, occurs and is
observed by an observer at rest in an inertial reference frame. The wording “in a
frame” means that the observer is at rest with respect to the origin of that frame.
The event’s location and time of occurrence can be specified by the four coordi-
nates (x, y, z, t). We would like to be able to transform these coordinates from those
of an observer in one inertial frame to those of another observer in a frame moving
with uniform relative velocity compared with the first frame.
Consider two inertial frames S and S9 (Fig. 39.2). The S9 frame moves with a con-
stant velocity v
along the common x and x9 axes, where v
is measured relative to S.
We assume the origins of S and S9 coincide at t 5 0 and an event occurs at point P in
space at some instant of time. For simplicity, we show the observer O in the S frame
and the observer O9 in the S9 frame as blue dots at the origins of their coordinate
frames in Figure 39.2, but that is not necessary: either observer could be at any
fixed location in his or her frame. Observer O describes the event with space–time
coordinates (x, y, z, t), whereas observer O9 in S9 uses the coordinates (x9, y9, z9,
t9) to describe the same event. Model the origin of S9 as a particle under constant
velocity relative to the origin of S. As we see from the geometry in Figure 39.2, the
relationships among these various coordinates can be written
x9 5 x 2 vt y9 5 y z9 5 z t9 5 t
These equations are the Galilean space–time transformation equations. Note that
time is assumed to be the same in both inertial frames. That is, within the frame-
work of classical mechanics, all clocks run at the same rate, regardless of their
velocity, so the time at which an event occurs for an observer in S is the same as the
time for the same event in S9. Consequently, the time interval between two succes-
sive events should be the same for both observers. Although this assumption may
seem obvious, it turns out to be incorrect in situations where v is comparable to the
speed of light.
Now suppose a particle moves through a displacement of magnitude dx along
the x axis in a time interval dt as measured by an observer in S. It follows from Equa-
tions 39.1 that the corresponding displacement dx9 measured by an observer in S9 is
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