Physics For Scientists And Engineers 6E - part 309

Questions

1233

diffraction grating consists of a large number of equally spaced, identical slits.

The condition for intensity maxima in the interference pattern of a diffraction grating
for normal incidence is

(38.10)

where d is the spacing between adjacent slits and m is the order number of the diffrac-
tion pattern. The resolving power of a diffraction grating in the mth order of the dif-
fraction pattern is

(38.12)

where N is the number of lines in the grating that are illuminated.

When polarized light of intensity I

max

is emitted by a polarizer and then incident

on an analyzer, the light transmitted through the analyzer has an intensity equal to
I

max

cos

, where ! is the angle between the polarizer and analyzer transmission axes.

In general, reflected light is partially polarized. However, reflected light is com-

pletely polarized when the angle of incidence is such that the angle between the
reflected and refracted beams is 90°. This angle of incidence, called the

polarizing

angle !

, satisfies

Brewster’s law:

(38.15)

where n is the index of refraction of the reflecting medium.

n " tan!

R " Nm

m " 0, #1, #2, #3,

$ $ $

d sin!

bright

Why can you hear around corners, but not see around
corners?

2. Holding your hand at arm’s length, you can readily block

sunlight from reaching your eyes. Why can you not block
sound from reaching your ears this way?

3. Observe the shadow of your book when it is held a few

inches above a table with a small lamp several feet above
the book. Why is the shadow somewhat fuzzy at the
edges?

4. Knowing that radio waves travel at the speed of light and

that  a  typical  AM  radio  frequency  is  1 000 kHz  while  an
FM radio  frequency  might  be  100 MHz,  estimate  the
wavelengths  of  typical  AM  and  FM  radio  signals.  Use  this
information to explain why AM radio stations can fade out
when you drive your car through a short tunnel or under-
pass, when FM radio stations do not.
Describe the change in width of the central maximum of
the single-slit diffraction pattern as the width of the slit is
made narrower.

6. John William Strutt, Lord Rayleigh (1842–1919), is

known as the last person to understand all of physics and
all  of  mathematics.  He  invented  an  improved  foghorn.
To  warn  ships  of  a  coastline,  a  foghorn  should  radiate
sound  in  a  wide  horizontal  sheet  over  the  ocean’s  sur-
face.  It  should  not  waste  energy  by  broadcasting  sound
upward.  It  should  not  emit  sound  downward,  because
the  water  in  front  of  the  foghorn  would  reflect  that
sound  upward.  Rayleigh’s  foghorn  trumpet  is  shown  in
Figure  Q38.6.  Is  it  installed  in  the  correct  orientation?
Decide  whether  the  long  dimension  of  the  rectangular
opening  should  be  horizontal  or  vertical,  and  argue  for
your decision.

7. Featured in the motion picture M*A*S*H (20th Century

Fox, Aspen Productions, 1970) is a loudspeaker mounted
on an exterior wall of an Army barracks. It has an approxi-
mately rectangular aperture. Its design can be thought of as
based on Lord Rayleigh’s foghorn trumpet, described in
Question 6. Borrow or rent a copy of the movie, sketch the
orientation of the loudspeaker, decide whether it is installed
in the correct orientation, and argue for your decision.

Q U E S T I O N S

Figure Q38.6

1234

C H A P T E R 3 8 • Diffraction Patterns and Polarization

8. Assuming that the headlights of a car are point sources,

estimate the maximum distance from an observer to the
car at which the headlights are distinguishable from each
other.

9. A laser beam is incident at a shallow angle on a machinist’s

ruler that has a finely calibrated scale. The engraved
rulings on the scale give rise to a diffraction pattern on a
screen. Discuss how you can use this technique to obtain a
measure of the wavelength of the laser light.

10. When you receive a chest x-ray at a hospital, the rays pass

through a series of parallel ribs in your chest. Do the ribs
act as a diffraction grating for x-rays?

11. Certain sunglasses use a polarizing material to reduce the

intensity of light reflected from shiny surfaces. What orien-
tation of polarization should the material have to be most
effective?

12. During the “day” on the Moon (when the Sun is visible),

you see a black sky and the stars can be clearly seen.
During the day on the Earth, you see a blue sky with no
stars. Account for this difference.

13. You can make the path of a light beam visible by placing

dust in the air (perhaps by clapping two blackboard
erasers in the path of the light beam). Explain why you can
see the beam under these circumstances. In general, when
is light visible?

14. Is light from the sky polarized? Why is it that clouds seen

through Polaroid glasses stand out in bold contrast to
the sky?

15. If a coin is glued to a glass sheet and this arrangement is

held in front of a laser beam, the projected shadow has dif-
fraction rings around its edge and a bright spot in the
center. How is this possible?

16. How could the index of refraction of a flat piece of dark

obsidian glass be determined?

17. A laser produces a beam a few millimeters wide, with uni-

form intensity across its width. A hair is stretched vertically
across the front of the laser to cross the beam. How is the
diffraction pattern it produces on a distant screen related
to that of a vertical slit equal in width to the hair? How
could you determine the width of the hair from measure-
ments of its diffraction pattern?

18. A radio station serves listeners in a city to the northeast of

its broadcast site. It broadcasts from three adjacent towers
on a mountain ridge, along a line running east and west.
Show  that  by  introducing  time  delays  among  the  signals
the individual towers radiate, the station can maximize net
intensity in the direction toward the city (and in the oppo-
site  direction)  and  minimize  the  signal  transmitted  in
other  directions.  The  towers  together  are  said  to  form  a
phased array.

Section 38.2 Diffraction Patterns from Narrow Slits

1. Helium–neon laser light (% " 632.8 nm) is sent through a

0.300-mm-wide single slit. What is the width of the central
maximum on a screen 1.00 m from the slit?

A beam of green light is diffracted by a slit of width
0.550 mm. The diffraction pattern forms on a wall 2.06 m
beyond the slit. The distance between the positions of zero
intensity on both sides of the central bright fringe is
4.10 mm. Calculate the wavelength of the laser light.

A screen is placed 50.0 cm from a single slit, which is

illuminated with 690-nm light. If the distance between the
first and third minima in the diffraction pattern is 3.00 mm,
what is the width of the slit?

4. Coherent microwaves of wavelength 5.00 cm enter a long,

narrow window in a building otherwise essentially opaque
to the microwaves. If the window is 36.0 cm wide, what is
the distance from the central maximum to the first-order
minimum along a wall 6.50 m from the window?

5. Sound with a frequency 650 Hz from a distant source passes

through a doorway 1.10 m wide in a sound-absorbing
wall. Find the number and approximate directions of

the diffraction-maximum beams radiated into the space
beyond.

6. Light of wavelength 587.5 nm illuminates a single slit

0.750 mm in width. (a) At what distance from the slit should
a screen be located if the first minimum in the diffraction
pattern is to be 0.850 mm from the center of the principal
maximum? (b) What is the width of the central maximum?

7. A beam of laser light of wavelength 632.8 nm has a circular

cross section 2.00 mm in diameter. A rectangular aperture
is to be placed in the center of the beam so that, when the
light  falls  perpendicularly  on  a  wall  4.50 m  away,  the
central  maximum  fills  a  rectangle  110 mm  wide  and
6.00 mm  high.  The  dimensions  are  measured  between
the minima  bracketing  the  central  maximum.  Find  the
required width and height of the aperture.

What If? Assume the light in Figure 38.5 strikes the single
slit at an angle ) from the perpendicular direction. Show
that Equation 38.1, the condition for destructive interfer-
ence, must be modified to read

sin !

dark

%
a

sin )

= straightforward, intermediate, challenging

= full solution available in the Student Solutions Manual and Study Guide

= coached solution with hints available at http://www.pse6.com

= computer useful in solving problem

= paired numerical and symbolic problems

P R O B L E M S

Problems

1235

A diffraction pattern is formed on a screen 120 cm away
from a 0.400-mm-wide slit. Monochromatic 546.1-nm light
is used. Calculate the fractional intensity I/I

max

at a point

on the screen 4.10 mm from the center of the principal
maximum.

10.

Coherent light of wavelength 501.5 nm is sent through two
parallel  slits  in  a  large  flat  wall.  Each  slit  is  0.700 /m  wide.
Their  centers  are  2.80 /m  apart.  The  light  then  falls  on  a
semicylindrical  screen,  with  its  axis  at  the  midline  between
the slits. (a) Predict the direction of each interference maxi-
mum  on  the  screen,  as  an  angle  away  from  the  bisector  of
the line joining the slits. (b) Describe the pattern of light on
the screen, specifying the number of bright fringes and the
location of each. (c) Find the intensity of light on the screen
at the center of each bright fringe, expressed as a fraction of
the light intensity I

max

at the center of the pattern.

Section 38.3 Resolution of Single-Slit and Circular

Apertures

11. The pupil of a cat’s eye narrows to a vertical slit of width

0.500 mm in daylight. What is the angular resolution for
horizontally separated mice? Assume that the average
wavelength of the light is 500 nm.

12. Find the radius a star image forms on the retina of the eye

if the aperture diameter (the pupil) at night is 0.700 cm
and the length of the eye is 3.00 cm. Assume the represen-
tative wavelength of starlight in the eye is 500 nm.

A helium–neon laser emits light that has a wavelength

of 632.8 nm. The circular aperture through which the
beam emerges has a diameter of 0.500 cm. Estimate the
diameter of the beam 10.0 km from the laser.

14. You are vacationing in a Wonderland populated by friendly

elves  and  a  cannibalistic  Cyclops  that  devours  physics
students.  The  elves  and  the  Cyclops  look  precisely  alike
(everyone  wears  loose  jeans  and  sweatshirts)  except  that
each  elf  has  two  eyes,  about  10.0 cm  apart,  and  the
Cyclops—you  guessed  it—has  only  one  eye  of  about  the
same  size  as  an  elf’s.  The  elves  and  the  Cyclops  are  con-
stantly at war with each other, so they rarely sleep and all
have  red  eyes,  predominantly  reflecting  light  with  a
wavelength of 660 nm. From what maximum distance can
you  distinguish  between  a  friendly  elf  and  the  predatory
Cyclops?  The  air  in  Wonderland  is  always  clear.  Dilated
with fear, your pupils have a diameter of 7.00 mm.

15.

Narrow, parallel, glowing gas-filled tubes in a variety of colors
form  block  letters  to  spell  out  the  name  of  a  night  club.
Adjacent  tubes  are  all  2.80 cm  apart.  The  tubes  forming
one letter  are  filled  with  neon  and  radiate  predominantly
red  light  with  a  wavelength  of  640 nm.  For  another  letter,
the tubes  emit  predominantly  violet  light  at  440 nm.  The
pupil  of  a  dark-adapted  viewer’s  eye  is  5.20 mm  in  diame-
ter. If she is in a certain range of distances away, the viewer
can  resolve  the  separate  tubes  of  one  color  but  not  the
other. Which color is easier to resolve? The viewer’s distance
must  be  in  what  range  for  her  to  resolve  the  tubes  of
only one color?

16. On the night of April 18, 1775, a signal was sent from the

steeple of Old North Church in Boston to Paul Revere,

13.

who  was  1.80 mi  away:  “One  if  by  land,  two  if  by  sea.”  At
what  minimum  separation  did  the  sexton  have  to  set  the
lanterns  for  Revere  to  receive  the  correct  message  about
the  approaching  British?  Assume  that  the  patriot’s  pupils
had  a  diameter  of  4.00 mm  at  night  and  that  the  lantern
light had a predominant wavelength of 580 nm.
The  Impressionist  painter  Georges  Seurat  created  paint-
ings  with  an  enormous  number  of  dots  of  pure  pigment,
each  of  which  was  approximately  2.00 mm  in  diameter.
The idea was to have colors such as red and green next to
each  other  to  form  a  scintillating  canvas  (Fig.  P38.17).
Outside  what  distance  would  one  be  unable  to  discern
individual  dots  on  the  canvas?  (Assume  that  % " 500 nm
and that the pupil diameter is 4.00 mm.)

17.

Figure P38.17 Sunday Afternoon on the Island of La Grande Jatte, by

Georges Seurat.

SuperStock

18. A binary star system in the constellation Orion has an angu-

lar interstellar separation of 1.00 & 10

rad. If % " 500 nm,

what is the smallest diameter the telescope can have to just
resolve the two stars?

19.

A  spy  satellite  can  consist  essentially  of  a  large-diameter
concave  mirror  forming  an  image  on  a  digital-camera
detector 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. Can a spy satellite read
a license plate? Can it read the date on a dime? Argue for
your  answers  by  making  an  order-of-magnitude  calcula-
tion, specifying the data you estimate.

20.

A circular radar antenna on a Coast Guard ship has a
diameter of 2.10 m and radiates at a frequency of 15.0 GHz.
Two small boats are located 9.00 km away from the ship.
How close together could the boats be and still be detected
as two objects?

21. Grote Reber was a pioneer in radio astronomy. He con-

structed a radio telescope with a 10.0-m-diameter receiving
dish. What was the telescope’s angular resolution for 2.00-m
radio waves?

22.

When Mars is nearest the Earth, the distance separating
the two planets is 88.6 & 10

km. Mars is viewed through a

telescope whose mirror has a diameter of 30.0 cm. (a) If
the wavelength of the light is 590 nm, what is the angular
resolution of the telescope? (b) What is the smallest
distance that can be resolved between two points on Mars?

1236

C H A P T E R 3 8 • Diffraction Patterns and Polarization

Section 38.4 The Diffraction Grating

White light is spread out into its spectral components by
a diffraction grating. If the grating has 2 000 grooves per
centimeter, at what angle does red light of wavelength
640 nm appear in first order?

24.

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 wavelength of the
laser light.

The hydrogen spectrum has a red line at 656 nm and

a blue line at 434 nm. What are the angular separations
between these two spectral lines obtained with a diffraction
grating that has 4 500 grooves/cm?

26. A helium–neon laser (% " 632.8 nm) is used to calibrate a

diffraction grating. If the first-order maximum occurs at
20.5°, what is the spacing between adjacent grooves in the
grating?

27. Three discrete spectral lines occur at angles of 10.09°,

13.71°, and 14.77° in the first-order spectrum of a grating
spectrometer. (a) If the grating has 3 660 slits/cm, what
are the wavelengths of the light? (b) At what angles are
these lines found in the second-order spectrum?

28.

Show that, whenever white light is passed through a diffrac-
tion grating of any spacing size, the violet end of the contin-
uous visible spectrum in third order always overlaps with red
light at the other end of the second-order spectrum.

A diffraction grating of width 4.00 cm has been ruled

with 3 000 grooves/cm. (a) What is the resolving power of
this grating in the first three orders? (b) If two monochro-
matic waves incident on this grating have a mean
wavelength of 400 nm, what is their wavelength separation
if they are just resolved in the third order?

30. The laser in a CD player must precisely follow the spiral

track,  along  which  the  distance  between  one  loop  of  the
spiral  and  the  next  is  only  about  1.25 /m.  A  feedback
mechanism  lets  the  player  know  if  the  laser  drifts  off  the
track,  so  that  the  player  can  steer  it  back  again.  Figure
P38.30 shows how a diffraction grating is used to provide
information  to  keep  the  beam  on  track.  The  laser  light
passes through a diffraction grating just before it reaches
the  disk.  The  strong  central  maximum  of  the  diffraction
pattern is used to read the information in the track of pits.
The two first-order side maxima are used for steering. The
grating  is  designed  so  that  the  first-order  maxima  fall  on
the  flat  surfaces  on  both  sides  of  the  information  track.
Both side beams are reflected into their own detectors. As
long as both beams are reflecting from smooth nonpitted
surfaces, they are detected with constant high intensity. If
the main beam wanders off the track, however, one of the
side  beams  will  begin  to  strike  pits  on  the  information
track and the reflected light will diminish. This change is
used with an electronic circuit to guide the beam back to

29.

25.

23.

Note: In the following problems, assume that the light is
incident normally on the gratings.

the  desired  location.  Assume  that  the  laser  light  has  a
wavelength  of  780 nm  and  that  the  diffraction  grating  is
positioned  6.90 /m  from  the  disk.  Assume  that  the
first-order beams are to fall on the disk 0.400 /m on either
side of the information track. What should be the number
of grooves per millimeter in the grating?
A  source  emits  531.62-nm  and  531.81-nm  light.  (a)  What
minimum number of grooves is required for a grating that
resolves  the  two  wavelengths  in  the  first-order  spectrum?
(b) Determine the slit spacing for a grating 1.32 cm wide
that has the required minimum number of grooves.

32. A diffraction grating has 4 200 rulings/cm. On a screen

2.00 m from the grating, it is found that for a particular
order m, the maxima corresponding to two closely spaced
wavelengths of sodium (589.0 nm and 589.6 nm) are sepa-
rated by 1.59 mm. Determine the value of m.
A grating with 250 grooves/mm is used with an incandes-
cent light source. Assume the visible spectrum to range in
wavelength from 400 to 700 nm. In how many orders can
one see (a) the entire visible spectrum and (b) the short-
wavelength region?

34.

A wide beam of laser light with a wavelength of 632.8 nm is
directed  through  several  narrow  parallel  slits,  separated  by
1.20 mm, and falls on a sheet of photographic film 1.40 m
away.  The  exposure  time  is  chosen  so  that  the  film  stays
unexposed everywhere except at the central region of each
bright fringe. (a) Find the distance between these interfer-
ence  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  trans-
parency  and  allowed  to  fall  on  a  screen  1.40 m  beyond.
(b) Argue that several narrow parallel bright regions, sepa-
rated by 1.20 mm, will appear on the screen, as real images
of the original slits. If at last the screen is removed, light will
diverge from the images of the original slits with the same
reconstructed  wave  fronts  as  the  original  slits  produced.
(Suggestion: You may find it useful to draw diagrams similar

33.

31.

Laser

Diffraction

grating

Central

maximum

First-order

maxima

Compact disc

Figure P38.30

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