chapter 22 problems

8/15/2019 Chapter 22 Problems

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PROBLEMS

1, 2, 3 = straightforward, intermediate, challenging = full solution available in Student Solutions

Manual/Study Guide = biomedical application

Section 22.1 The Nature of Light

1. During the Apollo X Moon landing, a retro!

reflecting panel was erected on the Moon"s

surface# $he speed of light may be found by

measuring the time it ta%es a laser beam to travel

from &arth, reflect from the panel, and return to

&arth# f this interval is measured to be '#() s,

what is the measured speed of light* $a%e the

center!to!center distance from &arth to the Moon

to be +#- . ) m# Assume the Moon is directlyoverhead and do not neglect the si0es of &arth

and the Moon#

2. 1igure 2''#' shows the apparatus used by

Armand 3# 4# 1i0eau 5))67)689 to measure

the speed of light# $he basic idea is to measure

the total time it ta%es light to travel from some

point to a distant mirror and bac%# f d is the

distance between the light source and the mirror

and if the transit time for one round trip is t , then

the speed of light is c = 'd /t # $o measure the

transit time, 1i0eau used a rotating toothed

wheel, which converts an otherwise continuous

beam of light to a series of light pulses# $he

rotation of the wheel controls what an observer

at the light source sees# 1or e:ample, assume

that the toothed wheel of the 1i0eau e:periment

has +8 teeth and is rotating at a speed of ';#(rev/s when the light from the source is

e:tinguished<that is, when a burst of light

passing through opening A in 1igure 2''#' is

bloc%ed by tooth B on return# f the distance to

the mirror is ; ( m, find the speed of light#

Figure P22.2

3. n an e:periment to measure the speed of lightusing the apparatus of 1i0eau described in the

preceding problem, the distance between light

source and mirror was ))#-( %m and the wheel

had ;' notches# $he e:perimentally determined

value of c was '#66 . ) m/s# alculate the

minimum angular speed of the wheel for this

e:periment#

4. Albert A# Michelson very carefully measuredthe speed of light using an alternative version of

the techni>ue developed by 1i0eau# 5See

2roblem '#9 1igure 2''#- shows the approach he

used# 4ight was reflected from one face of a

rotating eight!sided mirror toward a stationary

mirror +(# %m away# At certain rates of rotation,

the returning beam of light was directed toward

the eye of an observer as shown# 5a9 ?hat

minimum angular speed must the rotating mirror

have in order that side A will have rotated to

position @, causing the light to be reflected to

the eye* 5b9 ?hat is the ne:t!higher angular

velocity that will enable the source of light to be

seen*

Figure P22.4



5. 1igure 2''#( shows an apparatus used to

measure the speed distribution of gas molecules#

t consists of two slotted rotating dis%s separated

by a distance d , with the slots displaced by the

angle θ # Suppose the speed of light is measured by sending a light beam toward the left dis% of

this apparatus# 5a9 Show that a light beam will beseen in the detector 5that is, will ma%e it through

both slots9 only if its speed is given by c = d /θ ,

where is the angular speed of the dis%s and θ

is measured in radians# 5b9 ?hat is the measured

speed of light if the distance between the two

slotted rotating dis%s is '#( m, the slot in the

second dis% is displaced )/8 of one degree from

the slot in the first dis%, and the dis%s are

rotating at ( ((( rev/s*

Figure P22.5

Section 22.3 Reflection and Refraction

Section 22.4 The Law of Refraction

6. $he two mirrors in 1igure 2''#8 meet at a

right angle# $he beam of light in the vertical

plane P stri%es mirror ) as shown# 5a9 Determine

the distance the reflected light beam travels

before stri%ing mirror '# 5b9 n what direction

does the light beam travel after being reflected

from mirror '*

Figure P22.6

. An underwater scuba diver sees the Sun at an

apparent angle of -(#B from the vertical# ?hat

is the actual direction of the Sun*

!. 4ight is incident normally on a )#!cm layerof water that lies on top of a flat 4ucite plate

with a thic%ness of #( cm# 3ow much more

time is re>uired for light to pass through this

double layer than is re>uired to traverse the

same distance in air 5n4ucite = )#(69*

". A laser beam is incident at an angle of +#B

to the vertical onto a solution of corn syrup in

water# f the beam is refracted to )6#'-B to the

vertical, 5a9 what is the inde: of refraction of the

syrup solution* Suppose the light is red, with

vacuum wavelength 8+'# nm# 1ind its 5b9wavelength, 5c9 fre>uency, and 5d9 speed in the

solution#

1#. 1ind the speeds of light in 5a9 flint glass, 5b9

water, and 5c9 0ircon#

11. 4ight of wavelength C in vacuum has a

wavelength of -+ nm in water and a

wavelength of +6 nm in ben0ene# 5a9 ?hat is

the wavelength C of this light in vacuum* 5b9

sing only the given wavelengths, determine the

ratio of the inde: of refraction of ben0ene to thatof water#

12. 4ight of wavelength -+8 nm in air enters a

fishbowl filled with water, then e:its through the

crown!glass wall of the container# 1ind the

wavelengths of the light 5a9 in the water and 5b9in the glass#



13. A ray of light is incident on the surface of a

bloc% of clear ice at an angle of -#B with the

normal# 2art of the light is reflected and part is

refracted# 1ind the angle between the reflected

and refracted light#

14. A narrow beam of sodium yellow light 5C =(6 nm9 is incident from air on a smooth surface

of water at an angle of θ ) = +(#B# Determine the

angle of refraction θ ' and the wavelength of the

light in water#

15. A beam of light, traveling in air, stri%es the

surface of mineral oil at an angle of '+#)B with

the normal to the surface# f the light travels at

'#); = ) m/s through the oil, what is the angle

of refraction*

16. A flashlight on the bottom of a -#!m!deepswimming pool sends a ray upward and at an

angle so that the ray stri%es the surface of the

water '# m from the point directly above the

flashlight# ?hat angle 5in air9 does the emergingray ma%e with the water"s surface*

1. 3ow many times will the incident beam

shown in 1igure 2''#); be reflected by each of

the parallel mirrors*

Figure P22.1

1!. A ray of light stri%es a flat, '#!cm!thic%

bloc% of glass 5n = )#(9 at an angle of +#B

with the normal 51ig# 2''#)9# $race the light beam through the glass and find the angles of

incidence and refraction at each surface#

Figure P22.1!

1". ?hen the light ray in 2roblem ) passes

through the glass bloc%, it is shifted laterally bya distance d 51ig# 2''#)9# 1ind the value of d #

2#. 1ind the time re>uired for the light to pass

through the glass bloc% described in 2roblem )6#

21. $he light beam shown in 1igure 2''#')

ma%es an angle of '#B with the normal line

NN E in the linseed oil# Determine the angles θ

and θ E# 5$he refractive inde: for linseed oil is

)#-#9

Figure P22.21

22. A submarine is + m hori0ontally out from

the shore and ) m beneath the surface of the

water# A laser beam is sent from the sub so that it

stri%es the surface of the water at a point ') m

from the shore# f the beam Fust stri%es the top of

a building standing directly at the water"s edge,

find the height of the building#

23. $wo light pulses are emitted simultaneously

from a source# $he pulses ta%e parallel paths to a

detector 8#' m away, but one moves through air

and the other through a bloc% of ice# Determine



the difference in the pulses" times of arrival at

the detector#

24. A narrow beam of ultrasonic waves

reflects off the liver tumor in 1igure 2''#'-# f

the speed of the wave is )# less in the liver

than in the surrounding medium, determine the

depth of the tumor#

Figure P22.24

25. A beam of light both reflects and refracts at

the surface between air and glass as shown in

1igure 2''#'(# f the inde: of refraction of the

glass is n g , find the angle of incidence θ ) in the

air that would result in the reflected ray and the

refracted ray being perpendicular to each other#H Hint: Iemember the identity sin56B 7 θ 9 = cos

θ #J

Figure P22.25

26. $hree sheets of plastic have un%nown

indices of refraction# Sheet ) is placed on top of

sheet ', and a laser beam is directed onto the

sheets from above so that it stri%es the interface

at an angle of '8#(B with the normal# $he

refracted beam in sheet ' ma%es an angle of

+)#;B with the normal# $he e:periment is

repeated with sheet + on top of sheet ' and, with

the same angle of incidence, the refracted beam

ma%es an angle of +8#;B with the normal# f the

e:periment is repeated again with sheet ) on topof sheet +, what is the e:pected angle of

refraction in sheet +* Assume the same angle ofincidence#

2. An opa>ue cylindrical tan% with an open top

has a diameter of +# m and is completely filled

with water# ?hen the afternoon Sun reaches an

angle of '#B above the hori0on, sunlight ceases

to illuminate the bottom of the tan%# 3ow deep

is the tan%*

2!. A cylindrical cistern, constructed below

ground level, is +# m in diameter and '# m

deep and is filled to the brim with a li>uid whoseinde: of refraction is )#(# A small obFect rests on

the bottom of the cistern at its center# 3ow far

from the edge of the cistern can a girl whose

eyes are )#' m from the ground stand and stillsee the obFect*

Section 22.5 $i%&er%ion and Pri%'%

2". $he inde: of refraction for red light in water

is )#++), and that for blue light is )#+-# f a ray

of white light enters the water at an angle of

incidence of +#B, what are the underwaterangles of refraction for the blue and red

components of the light*

3#. A certain %ind of glass has an inde: of

refraction of )#8( for blue light of wavelength

-+ nm and an inde: of )#8)( for red light ofwavelength 8 nm# f a beam containing these

two colors is incident at an angle of +#B on a

piece of this glass, what is the angle between the

two beams inside the glass*

31. A ray of light stri%es the midpoint of oneface of an e>uiangular 58B!8B!8B9 glass prism

5n = )#(9 at an angle of incidence of +B# 5a9

$race the path of the light ray through the glass,

and find the angles of incidence and refraction at

each surface# 5b9 f a small fraction of light is

also reflected at each surface, find the angles of

reflection at these surfaces#



32. $he inde: of refraction for violet light in

silica flint glass is )#88 and that for red light is

)#8'# ?hat is the angular dispersion of visible

light passing through an e>uilateral prism of

ape: angle 8#B if the angle of incidence is(#B* 5See 1ig# 2''#+'#9

Figure P22.32

Section 22.! Total (nternal Reflection

33. alculate the critical angles for the following

materials when surrounded by airK 5a9 0ircon, 5b9

fluorite, 5c9 ice# Assume that C = (6 nm#

34. 1or (6!nm light, calculate the critical angle

for the following materials surrounded by airK5a9 diamond and 5b9 flint glass#

35. Iepeat 2roblem +- when the materials are

surrounded by water#

36. A beam of light is incident from air on the

surface of a li>uid# f the angle of incidence is

+#B and the angle of refraction is ''#B, find

the critical angle for the li>uid when surrounded

by air#

3. A light pipe consists of a central strand of

material surrounded by an outer coating# $he

interior portion of the pipe has an inde: of

refraction of )#8# f all rays stri%ing the interior

walls of the pipe with incident angles greater

than (6#(B are subFect to total internal reflection,what is the inde: of refraction of the coating*

3!. Determine the ma:imum angle θ for which

the light rays incident on the end of the light

pipe in 1igure 2''#+ are subFect to total internal

reflection along the walls of the pipe# Assume

that the light pipe has an inde: of refraction of

)#+8 and that the outside medium is air#

Figure P22.3!

3". onsider a common mirage formed by

super!heated air Fust above a roadway# A truc%

driver whose eyes are '# m above the road,

where n = )# +, loo%s forward# She has the

illusion of seeing a patch of water ahead on the

road, where her line of sight ma%es an angle of)#'B below the hori0ontal# 1ind the inde: of

refraction of the air Fust above the road surface#

5 Hint: $reat this as a problem in total internal

reflection#9

4#. A Fewel thief hides a diamond by placing iton the bottom of a public swimming pool# 3e

places a circular raft on the surface of the water

directly above and centered on the diamond, as

shown in 1igure 2''#-# f the surface of the

water is calm and the pool is '# m deep, find

the minimum diameter of the raft that would prevent the diamond from being seen#

Figure P22.4#

41. A room contains air in which the speed ofsound is +-+ m/s# $he walls of the room aremade of concrete, in which the speed of sound is

) ( m/s# 5a9 1ind the critical angle for total

internal reflection of sound at the concrete!air

boundary# 5b9 n which medium must the sound

be traveling in order to undergo total internal

reflection* 5c9 LA bare concrete wall is a highly



first day of spring, light from the rising Sun

enters a rectangular window in the eastern wall#

$he light traverses '#+; m hori0ontally to shine

perpendicularly on the wall opposite the

window# A prisoner observes the patch of lightmoving across this western wall and for the first

time forms his own understanding of the rotationof the &arth# 5a9 ?ith what speed does the

illuminated rectangle move* 5b9 $he prisoner

holds a small s>uare mirror flat against the wall

at one corner of the rectangle of light# $he

mirror reflects light bac% to a spot on the eastern

wall close beside the window# 3ow fast does the

smaller s>uare of light move across that wall*

5c9 Seen from a latitude of -#B north, the rising

Sun moves through the s%y along a line ma%ing

a (#B angle with the southeastern hori0on# n

what direction does the rectangular patch of

light on the western wall of the prisoner"s cellmove* 5d9 n what direction does the smaller

s>uare of light on the eastern wall move*

4". As shown in 1igure 2''#-6, a light ray isincident normally on one face of a +B!8B!6B

bloc% of dense flint glass 5a prism9 that is

immersed in water# 5a9 Determine the e:it angle

θ - of the ray# 5b9 A substance is dissolved in the

water to increase the inde: of refraction# At what

value of n' does total internal reflection cease at

point P *

Figure P22.4"

5#. A narrow beam of light is incident from air

onto a glass surface with inde: of refraction)#(8# 1ind the angle of incidence for which the

corresponding angle of refraction is one half theangle of incidence# 5 Hint: Oou might want to use

the trigonometric identity sin 'θ = ' sin θ cos θ #9

51. Nne techni>ue to measure the angle of a prism is shown in 1igure 2''#()# A parallel

beam of light is directed on the ape: of the

prism so that the beam reflects from opposite

faces of the prism# Show that the angular

separation of the two reflected beams is given by

B = ' A#

Figure P22.51

52. A -#!m!long pole stands vertically in a

la%e having a depth of '# m# ?hen the Sun is-#B above the hori0ontal, determine the length

of the pole"s shadow on the bottom of the la%e#

$a%e the inde: of refraction for water to be )#++#

53. A piece of wire is bent through an angle θ #

$he bent wire is partially submerged in ben0ene

5inde: of refraction = )#(9 so that to a person

loo%ing along the dry part, the wire appears to

be straight and ma%es an angle of +#B with the

hori0ontal# Determine the value of θ #

54. ?hen you loo% through a window, by howmuch time is the light you see delayed by having

to go through glass instead of air* Ma%e an

order!of!magnitude estimate on the basis of data

you specify# @y how many wavelengths is it

delayed*

55. A transparent cylinder of radius R = '# m

has a mirrored surface on its right half, as shown

in 1igure 2''#((# A light ray traveling in air is

incident on the left side of the cylinder# $heincident light ray and the e:iting light ray are

parallel and d = '# m# Determine the inde: ofrefraction of the material#



Figure P22.55

56. A laser beam stri%es one end of a slab ofmaterial, as in 1igure 2''#(8# $he inde: of

refraction of the slab is )#-# Determine the

number of internal reflections of the beam

before it emerges from the opposite end of theslab#

Figure P22.56

5. 1or this problem, refer to 1igure ''#)(# 1or

various angles of incidence, it can be shown that

the deviation angle δ is a minimum when the ray

passes through the glass so that the interior ray

is parallel to the base of the prism# A

measurement of this minimum angle of

deviation enables one to find the inde: ofrefraction of the prism material# Show that n is

given by the e:pression where A is the ape:

angle of the prism#

n =

+

'sin

9J5'

)Hsin min

A

A δ

5!. A hi%er stands on a mountain pea% near

sunset and observes a rainbow caused by water

droplets in the air about # %m away# $he

valley is '# %m below the mountain pea% andentirely flat# ?hat fraction of the complete

circular arc of the rainbow is visible to thehi%er*

5". A light ray is incident on a prism and

refracted at the first surface, as shown in 1igure

2''#(6# 4et φ represent the ape: angle of the

prism and n its inde: of refraction# 1ind, in

terms of n and φ, the smallest allowed value of

the angle of incidence at the first surface for

which the refracted ray will not undergo internal

reflection at the second surface#

Figure P22.5"

chapter 22 problems

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