Physics For Scientists And Engineers 6E - part 275

S E C T I O N 3 5 . 3 • The Ray Approximation in Geometric Optics

1097

Fizeau’s Method

The  first  successful  method  for  measuring  the  speed  of  light  by  means  of  purely
terrestrial techniques was developed in 1849 by French physicist Armand H. L. Fizeau
(1819–1896).  Figure  35.2  represents  a  simplified  diagram  of  Fizeau’s  apparatus.  The
basic  procedure  is  to  measure  the  total  time  interval  during  which  light  travels  from
some point to a distant mirror and back. If d is the distance between the light source
(considered to be at the location of the wheel) and the mirror and if the time interval
for one round trip is %t, then the speed of light is c ! 2d/%t.

To measure the transit time, Fizeau used a rotating toothed wheel, which converts a

continuous beam of light into a series of light pulses. The rotation of such a wheel
controls what an observer at the light source sees. For example, if the pulse traveling
toward the mirror and passing the opening at point A in Figure 35.2 should return to
the wheel at the instant tooth B had rotated into position to cover the return path, the
pulse would not reach the observer. At a greater rate of rotation, the opening at point
C could move into position to allow the reflected pulse to reach the observer. Knowing
the distance d, the number of teeth in the wheel, and the angular speed of the wheel,
Fizeau arrived at a value of 3.1 " 10

m/s. Similar measurements made by subsequent

investigators yielded more precise values for c, which led to the currently accepted
value of 2.997 9 " 10

m/s.

Toothed

wheel

Mirror

Figure 35.2 Fizeau’s method for

measuring the speed of light using

a rotating toothed wheel. The light

source is considered to be at the

location of the wheel; thus, the

distance d is known.

Example 35.1 Measuring the Speed of Light with Fizeau’s Wheel

Assume that Fizeau’s wheel has 360 teeth and is rotating at
27.5 rev/s when a pulse of light passing through opening A
in Figure 35.2 is blocked by tooth B on its return. If the
distance to the mirror is 7 500 m, what is the speed of light?

Solution The wheel has 360 teeth, and so it must have 360
openings. Therefore, because the light passes through open-
ing A but is blocked by the tooth immediately adjacent to A,
the wheel must rotate through an angular displacement of
(1/720) rev in the time interval during which the light pulse

makes its round trip. From the definition of angular speed,
that time interval is

Hence, the speed of light calculated from this data is

2.97 " 10

m/s

c !

2(7

500 m)

5.05 " 10

t !

(1/720)

rev

27.5 rev/s

5.05 " 10

Rays

Wave fronts

Figure 35.3 A plane wave

propagating to the right. Note that

the rays, which always point in the

direction of the wave propagation,

are straight lines perpendicular to

the wave fronts.

35.3 The Ray Approximation in Geometric Optics

The field of

geometric optics involves the study of the propagation of light, with the

assumption that light travels in a fixed direction in a straight line as it passes through a
uniform  medium  and  changes  its  direction  when  it  meets  the  surface  of  a  different
medium or if the optical properties of the medium are nonuniform in either space or
time. As we study geometric optics here and in Chapter 36, we use what is called the
ray  approximation. To  understand  this  approximation,  first  note  that  the  rays  of  a
given wave are straight lines perpendicular to the wave fronts as illustrated in Figure
35.3  for  a  plane  wave.  In  the  ray  approximation,  we  assume  that  a  wave  moving
through a medium travels in a straight line in the direction of its rays.

If the wave meets a barrier in which there is a circular opening whose diameter is

much larger than the wavelength, as in Figure 35.4a, the wave emerging from the open-
ing continues to move in a straight line (apart from some small edge effects); hence,
the ray approximation is valid. If the diameter of the opening is on the order of the
wavelength, as in Figure 35.4b, the waves spread out from the opening in all directions.
This effect is called diffraction and will be studied in Chapter 37. Finally, if the opening is
much smaller than the wavelength, the opening can be approximated as a point source
of waves (Fig. 35.4c). Similar effects are seen when waves encounter an opaque object of
dimension d. In this case, when ( )) d, the object casts a sharp shadow.

The ray approximation and the assumption that ( )) d are used in this chapter

and in Chapter 36, both of which deal with geometric optics. This approximation is
very good for the study of mirrors, lenses, prisms, and associated optical instruments,
such as telescopes, cameras, and eyeglasses.

35.4 Reflection

When  a  light  ray  traveling  in  one  medium  encounters  a  boundary  with  another
medium, part of the incident light is reflected. Figure 35.5a shows several rays of a
beam  of  light  incident  on  a  smooth,  mirror-like,  reflecting  surface.  The  reflected
rays  are  parallel  to  each  other,  as  indicated  in  the  figure.  The  direction  of  a
reflected ray is in the plane perpendicular to the reflecting surface that contains the

1098

C H A P T E R 3 5 • The Nature of Light and the Laws of Geometric Optics

Active Figure 35.4 A plane wave of wavelength

(

is incident on a barrier in which

there is an opening of diameter d. (a) When

(

))

d, the rays continue in a straight-line

path, and the ray approximation remains valid. (b) When

(

! d, the rays spread out

after passing through the opening. (c) When

(

d, the opening behaves as a point

source emitting spherical waves.

(c)

(a)

(b)

<< d

≈ d

>> d

At the Active Figures link

at http://www.pse6.com, you

can adjust the size of the

opening and observe the effect

on the waves passing through.

(b)

(a)

Figure 35.5 Schematic representation of (a) specular reflection, where the reflected

rays are all parallel to each other, and (b) diffuse reflection, where the reflected rays

travel in random directions. (c) and (d) Photographs of specular and diffuse reflection

using laser light.

Courtesy of Henry Leap and Jim Lehman

(c)

(d)

incident ray. Reflection of light from such a smooth surface is called

specular

reflection. If the reflecting surface is rough, as shown in Figure 35.5b, the surface
reflects the rays not as a parallel set but in various directions. Reflection from any
rough surface is known as

diffuse reflection. A surface behaves as a smooth surface

as long as the surface variations are much smaller than the wavelength of the
incident light.

The difference between these two kinds of reflection explains why it is more

difficult to see while driving on a rainy night. If the road is wet, the smooth surface of
the water specularly reflects most of your headlight beams away from your car (and
perhaps into the eyes of oncoming drivers). When the road is dry, its rough surface
diffusely reflects part of your headlight beam back toward you, allowing you to see the
highway more clearly. In this book, we concern ourselves only with specular reflection
and use the term reflection to mean specular reflection.

Consider a light ray traveling in air and incident at an angle on a flat, smooth

surface, as shown in Figure 35.6. The incident and reflected rays make angles &

and

, respectively, where the angles are measured between the normal and the rays. (The

normal is a line drawn perpendicular to the surface at the point where the incident ray
strikes the surface.) Experiments and theory show that

the angle of reflection equals

the angle of incidence:

(35.2)

This relationship is called the

law of reflection.

& +

S E C T I O N 3 5 . 4 • Reflection

1099

Quick Quiz 35.1

In the movies, you sometimes see an actor looking in a

mirror and you can see his face in the mirror. During the filming of this scene, what
does the actor see in the mirror? (a) his face (b) your face (c) the director’s face
(d) the movie camera (e) impossible to determine

Example 35.2 The Double-Reflected Light Ray

Interactive

Normal

Incident

ray

Reflected

ray

′

At the Active Figures link

at http://www.pse6.com, vary

the incident angle and see the

effect on the reflected ray.

▲

PITFALL PREVENTION

35.1 Subscript Notation

We  use  the  subscript  1  to  refer
to parameters  for  the  light  in
the initial  medium.  When  light
travels  from  one  medium  to
another, we use the subscript 2 for
the parameters associated with the
light  in  the  new  medium.  In  the
current  discussion,  the  light  stays
in  the  same  medium,  so  we  only
have to use the subscript 1.

Active Figure 35.6 According to the law of reflection,

. The incident ray, the reflected ray, and the normal all

lie in the same plane.

Law of reflection

Two mirrors make an angle of 120° with each other, as illus-
trated in Figure 35.7a. A ray is incident on mirror M

at an

angle of 65° to the normal. Find the direction of the ray
after it is reflected from mirror M

Solution Figure 35.7a helps conceptualize this situation. The
incoming ray reflects from the first mirror, and the reflected
ray  is  directed  toward  the  second  mirror.  Thus,  there  is  a
second reflection from this latter mirror. Because the interac-
tions  with  both  mirrors  are  simple  reflections,  we  categorize
this problem as one that will require the law of reflection and
some  geometry.  To  analyze  the  problem,  note  that  from  the

law of reflection, we know that the first reflected ray makes an
angle of 65° with the normal. Thus, this ray makes an angle of
90° # 65° ! 25° with the horizontal.

From the triangle made by the first reflected ray and the

two mirrors, we see that the first reflected ray makes an an-
gle of 35° with M

(because the sum of the interior angles of

any triangle is 180°). Therefore, this ray makes an angle of
55° with the normal to M

. From the law of reflection, the

second reflected ray makes an angle of 55° with the normal
to M

To finalize the problem, let us explore variations in the

angle between the mirrors as follows.

As discussed in the

What If? section of the preceding example, if the angle between

two mirrors is 90°, the reflected beam returns to the source parallel to its original path.
This phenomenon, called retroreflection, has many practical applications. If a third mirror
is  placed  perpendicular  to  the  first  two,  so  that  the  three  form  the  corner  of  a  cube,
retroreflection works in three dimensions. In 1969, a panel of many small reflectors was
placed on the Moon by the Apollo 11 astronauts (Fig. 35.8a). A laser beam from the Earth
is  reflected  directly  back  on  itself  and its transit  time  is  measured.  This  information  is
used to determine the distance to the Moon with an uncertainty of 15 cm. (Imagine how
difficult it would be to align a regular flat mirror so that the reflected laser beam would
hit  a particular  location  on  the  Earth!)  A  more  everyday  application  is  found  in
automobile taillights. Part of the plastic making up the taillight is formed into many tiny
cube corners (Fig. 35.8b) so that headlight beams from cars approaching from the rear
are  reflected  back  to  the  drivers.  Instead  of  cube  corners,  small  spherical  bumps  are
sometimes used (Fig. 35.8c). Tiny clear spheres are used in a coating material found on
many road signs. Due to retroreflection from these spheres, the stop sign in Figure 35.8d
appears much brighter than it would if it were simply a flat, shiny surface reflecting most
of the light hitting it away from the highway.

Another practical application of the law of reflection is the digital projection

of movies, television shows, and computer presentations. A digital projector makes use

1100

C H A P T E R 3 5 • The Nature of Light and the Laws of Geometric Optics

in Figure 35.7b, we see that

, -

2. - 2(90/ # 0) ! 180/

, !

2(0 # .)

The change in direction of the light ray is angle 1, which is
180° # ,:

1 !

180/ # , ! 180/ # 2(0 # .)

180/ # 2[0 # (90/ - 0 # &)]

360/ # 2&

Notice that 1 is not equal to &. For & ! 120°, we obtain 1 !
120°, which happens to be the same as the mirror angle. But
this is true only for this special angle between the mirrors.
For example, if & ! 90°, we obtain 1 ! 180°. In this case,
the light is reflected straight back to its origin.

Figure 35.7 (Example 35.2) (a) Mirrors

and M

make an angle of 120° with each

other. (b) The geometry for an arbitrary

mirror angle.

Investigate this reflection situation for various mirror angles at the Interactive Worked Example link at http://www.pse6.com.

120

(b)

(a)

°–φ

α β

What If?

If the incoming and outgoing rays in Figure 35.7a

are extended behind the mirror, they cross at an angle of 60°,
so that the overall change in direction of the light ray is 120°.
This  is  the  same  as  the  angle  between  the  mirrors.  What  if
the  angle  between  the  mirrors  is  changed?  Is  the  overall
change  in  the  direction  of  the  light  ray  always  equal  to  the
angle between the mirrors?

Answer Making  a  general  statement  based  on  one  data
point is always a dangerous practice! Let us investigate the
change  in  direction  for  a  general  situation.  Figure  35.7b
shows the mirrors at an arbitrary angle & and the incoming
light  ray  striking  the  mirror  at  an  arbitrary  angle  0 with
respect to the normal to the mirror surface. In accordance
with  the  law of reflection  and  the  sum  of  the  interior
angles of a triangle, the angle . is 180° # (90° # 0) # & !
90° - 0 # &.  Considering  the  triangle  highlighted  in  blue

Content .. 273 274 275 276 ..