( ------38-7 A ray of light in au' strikes a glass surface. Is there arange of angles for which total reflection occurs?

38-8 As shown in Table 38-1, diamond has a much largerrefractive index than glass. Is theTe a largeT or smalleTrange of angles for which total internal reflection occursfor diamond, than for glass? Does this have anything to dowith the fact that a real diamond has more sparkle than aglass imitation?

38-9 Light is usually observed to travel in straight lines,while radio waves seem to be able to bend around obsta-cles. If both are electromagnetic waves, why the differ-ence?38-10 Sunlight or starlight passing through the earth'satmosphere is always bent toward the vertical. Why? Doesthis mean that a star isn't really where it appears to be?

38-11 The sun or moon usually appears flattened justbefore it sets. Is this related to refraction in the earth'satmosphere, mentioned in Question 38-1O?38-12 A student claimed that, because of atmosphericrefraction (cf. Question 38-10), the sun can be seen after it

PROBLEMS 729

has set, and that the day is therefore longer than it wouldbe if the earth had no atmosphere. First, what does hemean by saying the sun can be seen after it has set? Sec-ond, comment on the validity of his conclusion.

38-13 Can sound waves be reflected? Refracted? Giveexamples. Does Huygens' principle apply to sound waves?

38-14 Why should the wavelength of light change, butnot its frequency, in passing from one mater:ial to another.

38-15 When light is incident on an interface between twomaterials; the angle of the refracted ray depends on thewavelength, but the angle of the reflected ray does not.Why should Ulls be?38-16 A room is completely lined with mirrors. A lightsource inside the room is turned on and then off. Does thelight continue to reflect forever') If not, what happensto it?

38-17 When light slows davin as it enters glass from vac-uum, and speeds up again as it emerges, does it gain andlose energy in the process? Momentum?

Problems"\ 38-1 What is the wavelength in meters, microns, nanom- ~rove that a ray of light reflected from a plane mi.r-

eters, and angstrom units of (a) soft x-rays 'of frequency ror rotates through an angle '21J when the mirror rotates2 X 1017 Hz? (b) green light of frequency 5.6 X 1014 Hz? through an angle e about an axis perpendicular to the

38-2 The visible spectrum includes a wavelength range' plane .of incidence.from about 400 nmi to about 700 nm. Express these wave- '~ A parallel beam of light is incident on a prism, aslengths in inches. shown in Fig. 38-16. Part of the light is reflected from one

"a-8-3 Assuming the radius of the earth's orhit to be face and part from another. Show that the angle e between92,900,000 mi, and taking the best value of the speed of the two reflected beams is twice the angle A between thelight, compute the time required for light to travel a dis- two reflecting surfaces.tance equal to the diameter of the earth's orbit. Comparewith Roemer's value of 22 min.

38-4 Fizeau's measurements of the speed of light werecontinued by Cornu, using Fizeau's ~aratus but with thedistance between mirrors increased to 22.9 km. One of thetoothed wheels used was 40 mm in diameter and had 180teeth. Find the angular velocity at which it should rotateso that light transmitted through one opening will returnthrough the next.

38-5 A ray of light trayeling with speed c leaves point 1 ofFig. 38-15 and is reflected to point 2. Show that the timerequired for the light to travel from 1 to 2 is (Yl sec e 1 +y~ sec e~)/c.

,""

Figure 38-16

i.-8 A parallel beam of light makes an angle of 300 withthe surface of a glass plate having a refractive index of 1.50.a) What is the angle between the refracted beam and the

surface of the glass? '

b) What should be the angle of incidence 0 with thisplate for the angle of refraction to be 0/2?

~?=>?.---~v~~

Ij

iII

; Figure 38-15

-==:=:: X--ITURE AND PROPAGATION OF LIGHT

- -~t strikes a glass plate at an angle of incidence of~ - of the beam being reflected and part refracted. It

ed that the reflected and refracted portions make_ ""gle of 90° with each other. What is the index of re-~on of the glass?

10 A ray of light is incident on a plane surface sepa-raring two transparent substances of indices 1.60 and lAO.Tbe angle of incidence is 30° and the ray originates in the

/r::edi~f higher index. Compute the angle of refraction.

~l !'.. parallel-sided prate of glass having a refractive./ mdex of 1.60 is held on the surface of water in a tank. A ray

coming from above makes an angle of incidence of 45 ° wjththe top surlace of the glass.

a) What angle does the ray make wjth the nonnal in thewater?

Figure 38-17

among all possible paths between two points, the oneactually taken):>y a ray of light is that for which thetime of trav~IS a milZffmum. (In fact,' there are somecases where the time is I;l. maximum rather than a min-imum.)

38-18 A point light source is 5 cm below a water-air sur-face. Compute the angles of refraction of rays from thesource making angles with the normal of 10°,20°,30°, and40°, and show these rays in a carefully drawn full-size dia-gram.

38-19 A glass cube in air has a refractive index of 1.50.Parallel rays of light enter the top obliquely and then

vacuum) in glass whose index at this wavelength is strike a side Of the cube. Is it possible for the rays to1.50? emerge from this side?

b) What is the wavelength of these waves in the glass? I~ point so~rce of lig~t is 20 cm below t~e su~face of38-14 A glass plate 3 mm thick, of index 1.50, is placed rabody of water. Fmd the. dian:eter of the largest CIrcle atbetween a point source of light of wavelength 600 nm (in the surface through whIch light can emerge from thevacuum) and a screen. The distance from source to screeD water. . \" ' , '. ....-----is 3 cm. How many waves are there between source and 38-21 The index of refraction of the prism shown in Fig.

v?iceen? 38-18 is 1.56. A ray of light enters the prism at point a and

. ~-15 Th.e speed of light of wavele~gth 6?6 nm in heavy f?llows in the prism the path ab, which is parallel to theflmt glass IS 1.60 X 108 m ·S-I. What IS the mdex of refrac-. lme cd.tion of this glass? '. a) Sketch carefully the path of the ray from a point out-38-16 Light of a certain frequency has a wavelength in side th~ prism .at the lef~, thro.ugh the glass, and outwater of 442 nm. What is the wavelength of this light when some distance mto the aIr agam.it passes into carbon disulfide?

..J.. 38-17 A ray of light goes from point A in a medium wherer the v~loc~ty of lig~t is. VI to point B in a medium wh~re thevelOCIty 18 V2, as m FIg. 38-17. . "

a) Show that the time required for the light to go n-pm Ato B is

b) How does this angle vary wjth the refractive index ofthe glass?

38-12 An inside corneT of a cube is lined with mirrors. Aray of light is reflected successively from each of threemutually perpendicular mirrors; show that its final direc-tion is always exactly opposite to its initial direction. Thisprinciple is used in tail-light lenses and reflecting highway

~- a) ~hat is the speed of light of wavelength 500 nm (in

hI sec 81 h2 sec 82t = + --=----"-VI V2

b) Take the derivative of t wjth respect to 81, noting that81 and 82 are functionally related by the fact thatl = hI tan 81 +h2 tan 82, Set this derivativee.qual tozero to show that this time reaches its minimum valuewhen n1 sin 81= n2 sin 82, This is an example of

. Fer:mat's principle of least time, which states that

c

Figure 38-18

b)

38-30 Use the result of Problem 38-29 to find the angle ofdeviation for a ray of light-passing symmetrically througha prism having tlu:.ee equal angles (A = 60°) and n = 1.50.

~ A certain glass has a refractive index of l.50 for redlight (700 nm) and l.52 for violet light (400 nm). If bothcolors pass through symmetrically, as described in Prob-lem 38-30, and if A = 60°, find the difference between theangles of deviation for the two colors, using tQe result of

Figure 38-20 Problem 38-29.

'" ::::e angle between the original and final di-.....~- ........=:::::arr.

~es are construction lines only.)

- ~:: is incident normally on the short face of a-::0= prism, as in Fig. 38-19. A drop of liquid is

we hypotenuse of the prism. If the index of the~ :;'.50, find the maximum index the liquid may have

_ ~ !ight is to be totally reflected. I, .

Figure 38-19

/08-23 A 45°_45°_90° prjsmi~s i=ersed in water. Whatis the minimum index of refraction the prism may have ifitis to totally reflect a ray incident normally on one of itsshorter faces? ; . .

38-2.4 The velocity of a sound Wave is 330m's-1 in airand 1320 m 'S-l in water. 0

a) What is the critical angle for a sound ,\lave incident onthe surface between air and water? .

b) Which medium has the higher "index of refraction" for

gJ0Ud? ' ,5 Light is incident at an angle CPl(as in, Fig. 38-20)

on the upper surface of a transparent plate, the surfaces ofthe plate being plane and parallel to each other.

a) Prove that <!>l= CP2'·

b) Show that this is true for any number of different par-allel pIa tes.

c) Prove that the lateral displacement d of the emergent.beam is given by the relation

d = t sin(cpl - <h') ,cos cPt'

where t is the thickness of the plate.

T) 1

jJ1

PROBL£MS 731

38-26 A parallel beam of light containing wavelengths Aand B is incident on the face of a triangular glass prismhaving a refracting angle of 60°. The indices of refractionare n.1 = lAO and nB = l.60. If beam A goes through theprism at minimUID deviation, find (a) the angle of emer-g,ce ~ach beam and (b) the angle of deviation of each.

~ A,ray of light is incident at an angle of 60° on one.surface of a glass plate 2 cm thick, of index l.50. The me-dium on either side of the plate is air. Find the transverse

(,displacement between the incident and emergent rays.

~¥-;-;- The prism of Fig. 9{l-21 has a refractive index of

l.4~, and the angles A an/]Q? Two light rays m and n are.mallei as they enter the prism. What is the angle betweenthem after they emerge?

Figure 38-21

~ght passes symmetrically through a prism havingapex angle A, as shown in Fig. ,38-22. Show that the angleof deviation 8 (the angle,between'the initial' and final clirec-tions of the ray) is giveri' by

. A + 8 '. A, ,~-'sm --2-' - =.n sm 2 .

Figure 38-22

748 IMAGES FORMED BY A SINGLE SURFACE

Questions--?:

~an a person see a real image by looking backward infinite series of reflections can be seen. Discuss this phe-along the direction from which the rays come? A virtual nomenon in terms of images. Why do the distant imagesimage? Can you tell by looking whether an image is real or appear d k ?virtual? Hov>'can the two be distinguished? 9- n observing fish in an aquarium filled with39-2 Why does a plane mirror reverse left and right but ater; one can see clearly only when looking nearly per-not top and bottom? pendicularly to the glass wall; objects viewed at an oblique39-3 For a spherical mir'ror, if s = f, then s' = oo,and the angle always appear blurred. Why? Do the fish have thelateral magnification m is infinite. Does this make sense? If same prob,m when looking at you?

~

o, w, at does it mean? 9- an an image for:medby one reflecting or refract-; 4 According to the discussion of the preceding chap- 1 surface serve as an object for a second reflection orter liaM ravs are reversible. Are the formulas in Table 39-1 refraction? Does it matter whether the first image is real or

,to ...

still valid if object and image are interchanged? What does virtual?reversibility imply with respect to the forms of the various 39-11 A concave mirror (sometimes surrounded by lights)for:mulas? is often used as an aid for applying cosmetics to the face.39-5 If a spherical mirror is immersed in water, does its Why is such a mirror always concave rather than convex?focal length change? What considerations deter:mine its radius of curvature?39-6 For what range of object positions does a concave 39-12 A student claimed that one can start a fire on aspherical mirror for:ma real image. What about a convex sunny day by use of the sun's rays and a concave mirrror.spherical mirror? How is this done? Is the concept of image relevant? Could

39 7 If . f h t h' film' I d t th 1 ~e do the me thing with a convex mirror?- a pIece 0 p 0 ograp lC ISP ace a e oca- --tion of a real image, the film will record theimage. Can this, . person loo.ks~t his.reflectio~ in the concave .sidebe done with a virtual image? How might one record a a shmy spoon. Is It TIghtSldeup or mverted? What If hevirtual image? looks in the convex side?

39-8 When a room has mirrors on two opposing walls, an

a) It1

b) Ia

39-1:conVIposit39-1frompoin1this'

a)

b)

c)

39-:

,a)

b)7ro~ms

~ What is the size of the smallest vertical plane mirror tance from the earth is 386,000km. Find the diameter ofin which an observer standing erect can see his full-length the image of the moon formed by a spherical concave tele-image? . I~,e mirror of focal length 4 m.=-\, -. j , ,

39-2 The image of a tree just covers the length of a 5-cm \[(~ A spheric.a~oncave shaving mirror has a radius ofplane mirror when the mirror is held 30 cm from the eye. curvature of'30 cri{) -The tree is 100m from the mirror. What.is its height? a) What is t'he-~;gnification when the face i@cm from'39-3 An object is placed between two mirrors arranged at the vertex of the mirror?right angles to each other. b) Where is the image?a) Locate all of the images of the object. 39-8 A concave spherical mirror has a radius of curvatureb) Draw the paths ofrays from the object to the eye of an of 10cm. Make a diagram of the mirror to scale, and show

[ , ~rver. rays incident on it parallel to the axis and at distances of 1,~ An object 1 cm high is 20 cm from the vertex of a 2, 3, 4, and 5 cm from the a~is. .using a pr~tractor, c~n-

concave spherical mirror whose radius of curvature is 50 cm. struct the reflected rays and mdicate the pomts at whichCompute the position and size of the image. Is it real or l~~ss the axis.

\rrtual?Erect or inverted? - !" ',' ,. ~~...:.'!':: object i~ 16cm from the center o! a silvered\\39-5 A concave mirror is to form an image of the filament spheTIcalglass C~~tmas tree or~ame~t 8 cm.m .dlameter.

of a hea~ght lam~ on a,screen. 4m fr?m the mirror. ~he /VJ:,:?:at are the pOSItionand magnlficatlOn of ItS nnage?filament IS5= hIgh, an3-_thennage ISto be 40cm high.~_uL __~ concave mirror of radius 5 cm has a radius ofa) What should be the radius of curvature ofthe mirror? curvature of 20cm. -b) How far in front of the vertex of the mirror should the a) What is its focal length?r filament be placed? . b) If the mirror is immersed in water (refractive index

~39-6 The diameter of the moon is 3480 Ian and its dis- 1.33),what is its focal length?

metra'

3S;wa:fre!wl1

39:hE!

ofiIar:

di'ali

J311

Al. IIT;

dI'

'iI

~i1 An object 2 cm high is pl~ced 5 cm away from aconcave sphencal mIrror havmg radius of curvature of20 cm.

a) Draw a principal-ray diagram showing' formation ofthe image.

b) Determine-the position, size, orientation, and natureof the image ..

39-12 Prove that'the image formed of a real object by aconvex mirror is always virtual, no matter what the object

~osit~~ If light striking a convex mirror does not divergefrom an object point but instead is converging toward apoint at a (negative) distance s to the right of the mirror,this pain t is called a L'irtual object.

a) For a convex mirror having radius of curvature 10 cm,for what range of virtual-object positions is a l'ealimage formed?

b) What is the orientation of the image?

c) Draw a principal-ray diagram showing formation ofsuch an image ..

39-14 A tank whose bottom is a mirror is filled with waterto a depth of 20 cm. A small object hangs motionless 8 cmunder the surface of the water. What is the apparent depthof its image when viewed at normal incidence?

. ;39-15 A ray of light in air makes an angle of incidence ofV 45 0 at the sur:face of a sheet of ice. The ray is refracted

within the ice at an angle of 300•

a) What is the critical angle for the ice?

b) A speck of dirt is embedded 2 cm below the surface ofthe ice. What is its apparent depth when viewed at

• n~al incidence?

~ .A microscope is focused on the upper surface of aglass plate. A second plate is then placed over the flrst. Inorder to focus on the bottom sur:face of the second plate,the microscope must be raised 1 mm. In order to focus onthe upper surface it must be raised 2 mm farther. Find theindex of refraction of the second plate. (Tills problem illus-trates one method of measu_ring index of refraction.)

39-17 A layer of ether (n = 1.36) 2 cm deep floats onwater (n = 1.33) 4 cm deep. What is the apparent distancefrom the ether surface to the bottom of the water layer,when viewed at normal incidence?

39-18 The end of a long glass rod 8 cm in diameter has ahemispherical surface 4 cm in radius. The l'efractive indexof the glass is 1.50. Determine each position of the image ifan object is placed on the a.xis of the rod at the followingdistances from its end:

a) infinitely far, b) 16 em, c) 4 cm.

~The rod of Problem 39-18 is immersed in a liquid.An object 60 cm from the end of the rod and on its axis is

- imaged at ;p;;int 100 cm inside the l'od. What is the refrac-tive index of the liquid?

39-20 What should be the index of refraction of a trans-

PROBLEMS 749

parent sphere in order that paraxial rays from an infinitelydistant object will be brought to a focus at the vertex oft~e s~ce opposite the point of incidence?

~ The left end of a long glass rod 10 em in diameter,of index 1.50, is ground and polished to a convex hemi-spherical surface of radius 5 cm. An object in the form ofan arrow 1 rom long, at right angles to the axis of the rod, islocated on the axis 20 cm to the left-of the vertex of theconvex surface. F:ind the positi9P 'and magnification of theimage of the arrow formed by paraxial rays incident on theconvex surface.

39-22 A transparent rod 40 cm long is cut flat at one endand rounded to a hemispherical surface of 12 cm radius atthe other end. A small object is embedded within the rodalong its axis and halfway between its ends. When viewedfrom the fiat end of the rod the apparent depth or-theobject is 12.5 cm. What is its apparent depth when viewedfrom the curved end?

~solid glass hemisphere having a radius of 10 cmand a refractive index of 1.50 is placed with its flat facedownward on a table. A parallel beam of light of circularcross sec_tion 1 cm in diameter travels directly down wardand enters the hemisphere along its diameter. Wh,at is thediameter of the circle of light formed on the table?

39-24 A small tropical fish is at the center of a sphericalfish bowl 30 cm in diameter. Find its apparent position andmagnification to an observer outside the bowl. The effectof the thin walls of the bowl may be neglected.

39-25 In Fig. 39-20a the first focal length f is seen to bethe value of s corresponding to s' = co; in (b) the secondfocaUength I' is the value of s' when s = co.

a) Prove that nln' = fll'.b) Prove that the general relation between object and

image distance~.is

(al

(bl Figure 39-20

___~""'i"~,,~~~~~~:;:~~~;~~

IIIJI11

!1II

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~ I

i~

II

IIl

An arrow 1 mm long, at right angles to the axis and 20 cmto the left of the first vertex, constitutes the object for thefirst surface.

a) What constitutes the object for the second 'surface?

b) What is the object distance for the second surface?

c) Is the object feal or virtual?

d) What is the position of the image formed by the sec-ond surface?

e) What is the height of the final image?

40-4 The same rod as in Problem 40-3 is now shortenedto a distance of 10 cm between its vertices, the curvaturesof its ends remaining the same.

a) What is the object distance for the second surface?

b) Is the object real or virtual?

c) What is the position of the image fOfmed by the sec-ond surface?

d Is the image real or virtual, erect or inverted, withrespect to the original object?

e) What is the height of the final image?

40-5 A glass rod of refractive index 1.50 is ground andpolished at both ends to hemispherical surfaces of 5 cmradius. When an object is placed on the axis of the rod,20 cm from one end, the final image is formed 40 cm from

J:.he ~ite end. What. is the length of th~ rod?

~~"~ ~s,olid glass sphere of radius R and index 1.50 issilvered over one hemisphere; as in Fig. 40-25. A small ob-ject is located on the axis of the sphere at a distance 2Rfrom the pole of the unsilvered hemisphere. Find: the posi-tion of the final image after all refractions and reflectionshave taken place.

s---------

Figure 40-25

40-7 A narrow b-eam of parallel rays enters a solid glasssphere in a radial direction. At what point outside thesphere are these rays brought to a focus? The radius of thesphere is 3 cm and its i.ndex is 1.50_

40-8 A glass plate 2 cm thick, of index 1.50, having planeparallel faces, is held with its faces horizontal and its lowerface 8 cm above a printed page. Find the position of theimage of the page, formed by rays making a small anglewith the normal to the plate.

40-9

PROBLEMS 769

a) Show that the equation

is that of an equilateral hyperbola having as asymp-totes the lines x = f and y = f.

b) Construct a graph with object distance s as abscissa,and image distance s' as ordinate for a lens of focallength f, and for object distances from 0 to 00.

c) On the same set of axes, construct a graph of magnifi-cation (ordinate) vs. object distance.

40-10 A converging lens has a focal length of 10 cm. Forobject distances of 30 cm, 20 cm, 15 cm, and 5 cm deter-mine

a) image position, :

b) magnification,

c) whether the image is real or viJ:tual, .-.

d) whether the image is erect or inverted. _l

40-11 Sketch the various possible thin lenses obtainableby combining two surfaces whose radii of curvature are, inabsolute magnitude, 10 cm and 20 cm. Which are converg-ing and which are diverging? Find the focal length of eachlens ~e of glass of index 1.50.

2 The radii of curvature of the surfaces of a thin lensare + 10 cm and +30 cm. The index is 1.50.

a) Compute the position and size of the image of an ob-ject in the form of an arrow 1 cm high, perpendicular -to the lens axis, 40 cm to the left of the lens.

b) A second similar lens is placed 160 cm to the right of _the first. Find the position of the final image.---------c) Same as (b), except the second lens is 40 cm to therigh t of the first.

d) Same as (c), except the second lens is diverging, offocal length -40 cI1}.

40-13 An object is placed 18 cm from a screen.

a) At what points between object and screen maya lensof 4 cm focal length be placed to obtain an image onthe screen?

b) What is the magnification of the image for these posi-tions of the lens?

~n object is imaged by a lens on a screen placed12 cm from the lens. When the lens is moved 2 cm fartherfrom the object, the screen must be moved 2 cm closer tothe object to refocus it. What is the focal length of thelens?

40-15 Three thin lenses, each of focal length 20 cm, aTealigned on a common axis and adjacent lenses are sepa-rared by 30 cm. Find the position of the image of a smallobject on the axis, 60 em to the left of the first lens.

40-16 An equiconvex thin lens made of glass of index 1.50has a focal length in air of 30 cm. The lens is sealed into anopening in one end of a tank filled with water (index =1.33). At the end of the tank opposite the lens is a planemirror, 80 cm distant from the lens. Find the position ofthe image for:med by the lens-water-mirror system. of a