1106 CHPTER 33 The Nture nd Propgtion of Light RIDGING PROLEM Reflection nd Refrction Figure 33.38 shows rectngulr glss block tht hs metl reflector on one fce nd wter on n djoining fce. light bem strikes the reflector s shown. You grdully increse the ngle u of the light bem. If u Ú 59.2, no light enters the wter. Wht is the speed of light in this glss? 33.38 Light bem Glss Wter SOLUTION GUIDE See MsteringPhysics study re for Video Tutor solution. u Reflector IDENTIFY nd SET UP 1. Speculr reflection occurs where the light ry in the glss strikes the reflector. If no light is to enter the wter, we require tht there be reflection only nd no refrction where this ry strikes the glss wter interfce tht is, there must be totl internl reflection. 2. The trget vrible is the speed of light v in the glss, which you cn determine from the index of refrction n of the glss. (Tble 33.1 gives the index of refrction of wter.) Write down the equtions you will use to find n nd v. EXECUTE 3. Use the figure to find the ngle of incidence of the ry t the glss wter interfce. 4. Use the result of step 3 to find n. 5. Use the result of step 4 to find v. EVLUTE 6. How does the speed of light in the glss compre to the speed in wter? Does this mke sense? Problems For instructor-ssigned homework, go to www.msteringphysics.com.,..,... : Problems of incresing difficulty. CP: Cumultive problems incorporting mteril from erlier chpters. CLC: Problems requiring clculus. IO: iosciences problems. DISCUSSION QUESTIONS Q33.1 Light requires bout 8 minutes to trvel from the sun to the erth. Is it delyed pprecibly by the erth s tmosphere? Explin. Q33.2 Sunlight or strlight pssing through the erth s tmosphere is lwys bent towrd the verticl. Why? Does this men tht str is not relly where it ppers to be? Explin. Q33.3 bem of light goes from one mteril into nother. On physicl grounds, explin why the wvelength chnges but the frequency nd period do not. Q33.4 student climed tht, becuse of tmospheric refrction (see Discussion Question Q33.2), the sun cn be seen fter it hs set nd tht the dy is therefore longer thn it would be if the erth hd no tmosphere. First, wht does she men by sying tht the sun cn be seen fter it hs set? Second, comment on the vlidity of her conclusion. Q33.5 When hot ir rises from rditor or heting duct, objects behind it pper to shimmer or wver. Wht cuses this? Q33.6 Devise strightforwrd experiments to mesure the speed of light in given glss using () Snell s lw; (b) totl internl reflection; (c) rewster s lw. Q33.7 Sometimes when looking t window, you see two reflected imges slightly displced from ech other. Wht cuses this? Q33.8 If you look up from underneth towrd the surfce of the wter in your qurium, you my see n upside-down reflection of your pet fish in the surfce of the wter. Explin how this cn hppen. Q33.9 ry of light in ir strikes glss surfce. Is there rnge of ngles for which totl reflection occurs? Explin. Q33.10 When light is incident on n interfce between two mterils, the ngle of the refrcted ry depends on the wvelength, but the ngle of the reflected ry does not. Why should this be? Q33.11 slesperson t brgin counter clims tht certin pir of sunglsses hs Polroid filters; you suspect tht the glsses re just tinted plstic. How could you find out for sure? Q33.12 Does it mke sense to tlk bout the polriztion of longitudinl wve, such s sound wve? Why or why not? Q33.13 How cn you determine the direction of the polrizing xis of single polrizer? Q33.14 It hs been proposed tht utomobile windshields nd hedlights should hve polrizing filters to reduce the glre of oncoming lights during night driving. Would this work? How should the polrizing xes be rrnged? Wht dvntges would this scheme hve? Wht disdvntges? Q33.15 When sheet of plstic food wrp is plced between two crossed polrizers, no light is trnsmitted. When the sheet is stretched in one direction, some light psses through the crossed polrizers. Wht is hppening? Q33.16 If you sit on the bech nd look t the ocen through Polroid sunglsses, the glsses help to reduce the glre from sunlight reflecting off the wter. ut if you lie on your side on the bech, there is little reduction in the glre. Explin why there is difference. Q33.17 When unpolrized light is incident on two crossed polrizers, no light is trnsmitted. student sserted tht if third polrizer is inserted between the other two, some trnsmission will occur. Does this mke sense? How cn dding third filter increse trnsmission?
Exercises 1107 Q33.18 For the old rbbit-er style TV ntenns, it s possible to lter the qulity of reception considerbly simply by chnging the orienttion of the ntenn. Why? Q33.19 In Fig. 33.32, since the light tht is scttered out of the incident bem is polrized, why is the trnsmitted bem not lso prtilly polrized? Q33.20 You re sunbthing in the lte fternoon when the sun is reltively low in the western sky. You re lying flt on your bck, looking stright up through Polroid sunglsses. To minimize the mount of sky light reching your eyes, how should you lie: with your feet pointing north, est, south, west, or in some other direction? Explin your resoning. Q33.21 Light scttered from blue sky is strongly polrized becuse of the nture of the scttering process described in Section 33.6. ut light scttered from white clouds is usully not polrized. Why not? Q33.22 tmospheric hze is due to wter droplets or smoke prticles ( smog ). Such hze reduces visibility by scttering light, so tht the light from distnt objects becomes rndomized nd imges become indistinct. Explin why visibility through hze cn be improved by wering red-tinted sunglsses, which filter out blue light. Q33.23 The explntion given in Section 33.6 for the color of the setting sun should pply eqully well to the rising sun, since sunlight trvels the sme distnce through the tmosphere to rech your eyes t either sunrise or sunset. Typiclly, however, sunsets re redder thn sunrises. Why? (Hint: Prticles of ll kinds in the tmosphere contribute to scttering.) Q33.24 Huygens s principle lso pplies to sound wves. During the dy, the temperture of the tmosphere decreses with incresing ltitude bove the ground. ut t night, when the ground cools, there is lyer of ir just bove the surfce in which the temperture increses with ltitude. Use this to explin why sound wves from distnt sources cn be herd more clerly t night thn in the dytime. (Hint: The speed of sound increses with incresing temperture. Use the ides displyed in Fig. 33.37 for light.) Q33.25 Cn wter wves be reflected nd refrcted? Give exmples. Does Huygens s principle pply to wter wves? Explin. EXERCISES Section 33.2 Reflection nd Refrction 33.1. Two plne mirrors intersect t right ngles. lser bem strikes the first of them t point 11.5 cm from their point of intersection, s shown in Fig. E33.1 For wht ngle of incidence t the first mirror will this ry strike the midpoint of the second mirror (which is 28.0 cm long) fter reflecting from the first mirror? Figure E33.1 11.5 cm 33.2. IO Light Inside the Eye. The vitreous humor, trnsprent, geltinous fluid tht fills most of the eyebll, hs n index of refrction of 1.34. Visible light rnges in wvelength from 380 nm (violet) to 750 nm (red), s mesured in ir. This light trvels through the vitreous humor nd strikes the rods nd cones t the surfce of the retin. Wht re the rnges of () the wvelength, (b) the frequency, nd (c) the speed of the light just s it pproches the retin within the vitreous humor? 28.0 cm 33.3. bem of light hs wvelength of 650 nm in vcuum. () Wht is the speed of this light in liquid whose index of refrction t this wvelength is 1.47? (b) Wht is the wvelength of these wves in the liquid? 33.4. Light with frequency of 5.80 * 10 14 Hz trvels in block of glss tht hs n index of refrction of 1.52. Wht is the wvelength of the light () in vcuum nd (b) in the glss? 33.5. light bem trvels t 1.94 * 10 8 m>s in qurtz. The wvelength of the light in qurtz is 355 nm. () Wht is the index of refrction of qurtz t this wvelength? (b) If this sme light trvels through ir, wht is its wvelength there? 33.6.. Light of certin frequency hs wvelength of 438 nm in wter. Wht is the wvelength of this light in benzene? 33.7.. prllel bem of light in ir mkes n ngle of 47.5 with the surfce of glss plte hving refrctive index of 1.66. () Wht is the ngle between the reflected prt of the bem nd the surfce of the glss? (b) Wht is the ngle between the refrcted bem nd the surfce of the glss? 33.8.. lser bem shines long the surfce of block of trnsprent mteril (see Fig. E33.8.). Hlf of the bem goes stright to detector, while the other hlf trvels through the block nd then hits the detector. The time dely between the rrivl of the two light bems t the detector is 6.25 ns. Wht is the index of refrction of this mteril? 33.9. Light trveling in ir is incident on the surfce of block of plstic t n ngle of 62.7 to the norml nd is bent so tht it mkes 48.1 ngle with the norml in the plstic. Find the speed of light in the plstic. 33.10. () tnk contining methnol hs wlls 2.50 cm thick mde of glss of refrctive index 1.550. Light from the outside ir strikes the glss t 41.3 ngle with the norml to the glss. Find the ngle the light mkes with the norml in the methnol. (b) The tnk is emptied nd refilled with n unknown liquid. If light incident t the sme ngle s in prt () enters the liquid in the tnk t n ngle of 20.2 from the norml, wht is the refrctive index of the unknown liquid? 33.11.. s shown in Fig. E33.11, lyer of wter covers slb of mteril X in beker. ry of light trveling upwrd follows the pth indicted. Using the informtion on the figure, find () the index of refrction of mteril X nd (b) the ngle the light mkes with the norml in the ir. 33.12.. horizontl, prllel-sided plte of glss hving refrctive index of 1.52 is in contct with the surfce of Figure E33.8 n 5? 2.50 m Figure E33.11 Detector wter in tnk. ry coming from bove in ir mkes n ngle of incidence of 35.0 with the norml to the top surfce of the glss. () Wht ngle does the ry refrcted into the wter mke with the norml to the surfce? (b) Wht is the dependence of this ngle on the refrctive index of the glss? 33.13.. In mteril hving n index of refrction n, light ry hs frequency ƒ, wvelength l, nd speed v. Wht re the frequency, wvelength, nd speed of this light () in vcuum nd ir Wter 65 48 X
1108 CHPTER 33 The Nture nd Propgtion of Light (b) in mteril hving refrctive index n? In ech cse, express lrgest vlue the ngle cn Figure E33.22 your nswers in terms of only ƒ, l, v, n, nd n. hve without ny light refrcted 33.14. ry of light trveling in wter is incident on n interfce out of the prism t fce C if () with flt piece of glss. The wvelength of the light in the wter is 726 nm nd its wvelength in the glss is 544 nm. If the ry in the prism is immersed in ir nd (b) the prism is immersed in Incident wter mkes n ngle of 42.0 o with respect to the norml to the wter. ry interfce, wht ngle does the refrcted ry in the glss mke with 33.23. piece of glss with C respect to the norml? flt surfce is t the bottom of tnk of wter. If ry of light 33.15. ry of light is incident on plne surfce seprting trveling in the glss is incident on the interfce with the wter t two sheets of glss with refrctive indexes 1.70 nd 1.58. The ngle of incidence is 62.0, nd the ry origintes in the glss with n = 1.70. Compute the ngle of refrction. n ngle with respect to the norml tht is greter thn 62.0 o, no light is refrcted into the wter. For smller ngles of incidence, prt of the ry is refrcted into the wter. If the light hs wvelength 408 nm in the glss, wht is the wvelength of the light in the wter? 33.24.. We define the index of refrction of mteril for sound wves to be the rtio of the speed of sound in ir to the speed of sound in the mteril. Snell s lw then pplies to the refrction of Section 33.3 Totl Internl Reflection 33.16. flt piece of glss covers the top of verticl cylinder tht is completely filled with wter. If ry of light trveling in the glss is incident on the interfce with the wter t n ngle of u = 36.2 o, the ry refrcted into the wter mkes n ngle of 49.8 o with the norml to the interfce. Wht is the smllest vlue of the incident ngle u for which none of the ry refrcts into the wter? 33.17. Light Pipe. Light Figure E33.17 enters solid pipe mde of plstic hving n index of u refrction of 1.60. The light trvels prllel to the upper prt of the pipe (Fig. E33.17). You wnt to cut the fce so tht ll the light will reflect bck into the pipe fter it first strikes tht fce. () Wht is the lrgest tht u cn be if the pipe is in ir? (b) If the pipe is immersed in wter of refrctive index 1.33, wht is the lrgest tht u cn be? 33.18. bem of light is trveling inside solid glss cube hving index of refrction 1.53. It strikes the surfce of the cube from the inside. () If the cube is in ir, t wht minimum ngle with the norml inside the glss will this light not enter the ir t this surfce? (b) Wht would be the minimum ngle in prt () if the cube were immersed in wter? 33.19.. The criticl ngle for totl internl reflection t liquid ir interfce is 42.5. () If ry of light trveling in the liquid hs n ngle of incidence t the interfce of 35.0, wht ngle does the refrcted ry in the ir mke with the norml? (b) If ry of light trveling in ir hs n ngle of incidence t the interfce of 35.0, wht ngle does the refrcted ry in the liquid mke with the norml? 33.20. t the very end of Wgner s series of opers Ring of the Nibelung, rünnhilde tkes the golden ring from the finger of the ded Siegfried nd throws it into the Rhine, where it sinks to the bottom of the river. ssuming tht the ring is smll enough compred to the depth of the river to be treted s point nd tht the Rhine is 10.0 m deep where the ring goes in, wht is the re of the lrgest circle t the surfce of the wter over which light from the ring could escpe from the wter? 33.21. ry of light is trveling in glss cube tht is totlly immersed in wter. You find tht if the ry is incident on the glss wter interfce t n ngle to the norml lrger thn 48.7, no light is refrcted into the wter. Wht is the refrctive index of the glss? 33.22. Light is incident long the norml on fce of glss prism of refrctive index 1.52, s shown in Fig. E33.22. Find the sound wves. The speed of sound wve is 344 m> s in ir nd 1320 m> s in wter. () Which medium hs the higher index of refrction for sound? (b) Wht is the criticl ngle for sound wve incident on the surfce between ir nd wter? (c) For totl internl reflection to occur, must the sound wve be trveling in the ir or in the wter? (d) Use your results to explin why it is possible to her people on the opposite shore of river or smll lke extremely clerly. Section 33.4 Dispersion 33.25.. nrrow bem Figure E33.25 of white light strikes one fce of slb of silicte White light flint glss. The light is trveling prllel to the two 55.0 djoining fces, s shown in Fig. E33.25. For the trnsmitted light inside the glss, through wht ngle Silicte flint glss Du 5? u is the portion of the visible spectrum between 400 nm nd 700 nm dispersed? (Consult the grph in Fig. 33.18.) 33.26. bem of light strikes sheet of glss t n ngle of 57.0 with the norml in ir. You observe tht red light mkes n ngle of 38.1 with the norml in the glss, while violet light mkes 36.7 ngle. () Wht re the indexes of refrction of this glss for these colors of light? (b) Wht re the speeds of red nd violet light in the glss? Section 33.5 Polriztion 33.27. Unpolrized light with intensity I 0 is incident on two polrizing filters. The xis of the first filter mkes n ngle of 60.0 o with the verticl, nd the xis of the second filter is horizontl. Wht is the intensity of the light fter it hs pssed through the second filter? 33.28.. () t wht ngle bove the horizontl is the sun if sunlight reflected from the surfce of clm lke is completely polrized? (b) Wht is the plne of the electric-field vector in the reflected light? 33.29.. bem of unpolrized light of intensity I 0 psses through series of idel polrizing filters with their polrizing directions turned to vrious ngles s shown in Fig. E33.29. () Wht is the light intensity (in terms of I 0 ) t points,, nd C? (b) If we remove the middle filter, wht will be the light intensity t point C?
Problems 1109 Figure E33.29 I 0 Unpolrized 33.30. Light trveling in wter strikes glss plte t n ngle of incidence of 53.0 ; prt of the bem is reflected nd prt is refrcted. If the reflected nd refrcted portions mke n ngle of 90.0 with ech other, wht is the index of refrction of the glss? 33.31. prllel bem of unpolrized light in ir is incident t n ngle of 54.5 (with respect to the norml) on plne glss surfce. The reflected bem is completely linerly polrized. () Wht is the refrctive index of the glss? (b) Wht is the ngle of refrction of the trnsmitted bem? 33.32.. Light of originl intensity I 0 psses through two idel polrizing filters hving their polrizing xes oriented s shown in Fig. E33.32. You wnt to djust the ngle f so tht the intensity t point P is equl to I 0 >10. () If the originl light is unpolrized, wht should f be? (b) If the originl light is linerly polrized in the sme direction s the polrizing xis of the first polrizer the light reches, wht should f be? Figure E33.32 I 0 33.33.. bem of polrized light psses through polrizing filter. When the ngle between the polrizing xis of the filter nd the direction of polriztion of the light is u, the intensity of the emerging bem is I. If you now wnt the intensity to be I>2, wht should be the ngle (in terms of u) between the polrizing ngle of the filter nd the originl direction of polriztion of the light? 33.34. The refrctive index of certin glss is 1.66. For wht incident ngle is light reflected from the surfce of this glss completely polrized if the glss is immersed in () ir nd (b) wter? 33.35.. Unpolrized light of intensity 20.0 W>cm 2 is incident on two polrizing filters. The xis of the first filter is t n ngle of 25.0 counterclockwise from the verticl (viewed in the direction the light is trveling), nd the xis of the second filter is t 62.0 counterclockwise from the verticl. Wht is the intensity of the light fter it hs pssed through the second polrizer? 33.36... Three polrizing filters re stcked, with the polrizing xis of the second nd third filters t 23.0 nd 62.0, respectively, to tht of the first. If unpolrized light is incident on the stck, the light hs intensity 75.0 W>cm 2 fter it psses through the stck. If the incident intensity is kept constnt, wht is the intensity of the light fter it hs pssed through the stck if the second polrizer is removed? 33.37. Three Polrizing Filters. Three polrizing filters re stcked with the polrizing xes of the second nd third t 45.0 nd 90.0, respectively, with tht of the first. () If unpolrized light of intensity is incident on the stck, find the intensity nd I 0 60 f 90 P C stte of polriztion of light emerging from ech filter. (b) If the second filter is removed, wht is the intensity of the light emerging from ech remining filter? Section 33.6 Scttering of Light 33.38. bem of white light psses through uniform thickness of ir. If the intensity of the scttered light in the middle of the green prt of the visible spectrum is I, find the intensity (in terms of I) of scttered light in the middle of () the red prt of the spectrum nd (b) the violet prt of the spectrum. Consult Tble 32.1. PROLEMS 33.39.. The Corner Reflector. n inside corner of cube is lined with mirrors to mke corner reflector (see Exmple 33.3 in Section 33.2). ry of light is reflected successively from ech of three mutully perpendiculr mirrors; show tht its finl direction is lwys exctly opposite to its initil direction. 33.40. light bem is directed prllel to the xis of hollow cylindricl tube. When the tube contins only ir, it tkes the light 8.72 ns to trvel the length of the tube, but when the tube is filled with trnsprent jelly, it tkes the light 2.04 ns longer to trvel its length. Wht is the refrctive index of this jelly? 33.41.. IO Hert Sonogrm. Physicins use high-frequency 1ƒ = 1-5 MHz2 sound wves, clled ultrsound, to imge internl orgns. The speed of these ultrsound wves is 1480 m> s in muscle nd 344 m> s in ir. We define the index of refrction of mteril for sound wves to be the rtio of the speed of sound in ir to the speed of sound in the mteril. Snell s lw then pplies to the refrction of sound wves. () t wht ngle from the norml does n ultrsound bem enter the hert if it leves the lungs t n ngle of 9.73 from the norml to the hert wll? (ssume tht the speed of sound in the lungs is 344 m> s.) (b) Wht is the criticl ngle for sound wves in ir incident on muscle? 33.42... In physics lb, light with wvelength 490 nm trvels in ir from lser to photocell in 17.0 ns. When slb of glss 0.840 m thick is plced in the light bem, with the bem incident long the norml to the prllel fces of the slb, it tkes the light 21.2 ns to trvel from the lser to the photocell. Wht is the wvelength of the light in the glss? 33.43.. ry of light is incident in ir on block of trnsprent solid whose index of Figure P33.43 refrction is n. If n = 1.38, wht is the u lrgest ngle of incidence u for which totl internl reflection will occur t the verticl fce (point shown in Fig. P33.43)? 33.44. light ry in ir strikes the rightngle prism shown in Fig. P33.44. The prism ngle t is 30.0 o. This ry consists of two different wvelengths. When it emerges t fce, it hs been split into two different rys tht diverge from ech other by 8.50. Find the index of refrction Figure P33.44 of the prism for ech of the two wvelengths. Incident ry 33.45.. ry of light trveling in block of glss 12.0 1n = 1.522 is incident on the top surfce t n ngle of 57.2 with respect to the norml in the glss. If lyer of oil is plced on the top surfce 30.0 8.50
1110 CHPTER 33 The Nture nd Propgtion of Light of the glss, the ry is totlly reflected. Wht is the mximum possible index of refrction of the oil? 33.46... glss plte 2.50 mm thick, with n index of refrction of 1.40, is plced between point source of light with wvelength 540 nm (in vcuum) nd screen. The distnce from source to screen is 1.80 cm. How mny wvelengths re there between the source nd the screen? 33.47.. Old photogrphic pltes were mde of glss with lightsensitive emulsion on the front surfce. This emulsion ws somewht trnsprent. When bright point source is focused on the front of the plte, the developed photogrph will show hlo round the imge of the spot. If the glss plte is 3.10 mm thick nd the hlos hve n inner rdius of 5.34 mm, wht is the index of refrction of the glss? (Hint: Light from the spot on the front surfce is scttered in ll directions by the emulsion. Some of it is then totlly reflected t the bck surfce of the plte nd returns to the front surfce.) 33.48. fter long dy of driving you tke lte-night swim in motel swimming pool. When you go to your room, you relize tht you hve lost your room key in the pool. You borrow powerful flshlight nd wlk round the pool, shining the light into it. The light shines on the key, which is lying on the bottom of the pool, when the flshlight is held 1.2 m bove the wter surfce nd is directed t the surfce horizontl distnce of 1.5 m from the edge (Fig. P33.48). If the wter here is 4.0 m deep, how fr is the key from the edge of the pool? Figure P33.48 1.5 m 33.49. You sight long the rim of glss with verticl sides so tht the top rim is lined up with the opposite edge of the bottom (Fig. P33.49). The glss is thin-wlled, hollow cylinder 16.0 cm high. The dimeter of the top nd bottom of the glss is 8.0 cm. While you keep your eye in the sme position, friend fills the glss with trnsprent liquid, nd you then see dime tht is lying t the center of the bottom of the glss (Fig. P33.49b). Wht is the index of refrction of the liquid? 1.2 m? () 16.0 cm 8.0 cm 4.0 m Figure P33.49 (b) 33.50.. 45-45 -90 prism is immersed in wter. ry of light is incident normlly on one of its shorter fces. Wht is the minimum index of refrction tht the prism must hve if this ry is to be totlly reflected within the glss t the long fce of the prism? 33.51. thin lyer of ice 1n = 1.3092 flots on the surfce of wter 1n = 1.3332 in bucket. ry of light from the bottom of the bucket trvels upwrd through the wter. () Wht is the lrgest ngle with respect to the norml tht the ry cn mke t the ice wter interfce nd still pss out into the ir bove the ice? (b) Wht is this ngle fter the ice melts? 33.52.. Light is incident normlly on the short fce of 30 - Figure P33.52 60-90 prism (Fig. P33.52). drop of liquid is plced on the 60 30 hypotenuse of the prism. If the index of refrction of the prism is 1.62, find the mximum index 90 tht the liquid my hve if the light is to be totlly reflected. 33.53.. The prism shown in Fig. P33.53 Figure P33.53 hs refrctive index of 1.66, nd the ngles re 25.0. Two light rys m nd n re prllel s they enter the prism. Wht is the ngle between them fter they emerge? 33.54.. m horizontl cylindricl tnk 2.20 m in dimeter is hlf full of wter. The spce bove the wter is filled with pressurized gs of unknown refrctive index. n smll lser cn move long the curved bottom of the wter nd ims light bem towrd the center of the wter surfce (Fig. P33.54). You observe tht when the lser hs moved distnce Figure P33.54 S = 1.09 m or more (mesured long the curved surfce) from the lowest point in the wter, no light enters the gs. () Wht is Gs the index of refrction of the gs? (b) Wht minimum time does it tke the light bem to Lser trvel from the lser to the rim of the tnk when (i) S 7 1.09 m nd (ii) S 6 1.09 m? 33.55.. S When the sun is either rising or setting nd ppers to be just on the horizon, it is in fct below the horizon. The explntion for this seeming prdox is tht light from the sun bends slightly when entering the erth s tmosphere, s shown in Fig. P33.55. Since our perception is bsed on the ide tht light trvels in stright lines, we perceive the light to be coming from n pprent position tht is n ngle d bove the sun s true position. () Mke the simplifying ssumptions tht the tmosphere hs uniform density, nd hence uniform index of refrction n, nd extends to height h bove the erth s surfce, t which point it bruptly stops. Show tht the ngle d is given by d = rcsin nr R + h b - rcsin R R + h b where R = 6378 km is the rdius of the erth. (b) Clculte d using n = 1.0003 nd h = 20 km. How does this compre to the ngulr rdius of the sun, which is bout one qurter of degree?
Problems 1111 (In ctulity light ry from the sun bends grdully, not bruptly, since the density nd refrctive index of the tmosphere chnge grdully with ltitude.) Figure P33.55 h R R 1 h Erth tmosphere pprent position of the sun From the sun 33.56.. CLC Fermt s Principle of Lest Time. ry of light trveling with speed c leves point 1 shown in Fig. P33.56 nd is reflected to point 2. The ry strikes the reflecting surfce horizontl distnce x from point 1. () Show tht the time t required for the light to trvel from 1 to 2 is t = 2y 2 1 + x 2 + 2y 2 2 + 1l - x2 2 c (b) Tke the derivtive of t with respect to x. Set the derivtive equl to zero to show tht this time reches its minimum vlue when u 1 = u 2, which is the lw of reflection nd corresponds to the ctul pth tken by the light. This is n exmple of Fermt s principle of lest time, which sttes tht mong ll possible pths between two points, the one ctully tken by ry of light is tht for which the time of trvel is minimum. (In fct, there re some cses in which the time is mximum rther thn minimum.) d t = 2h 2 1 + x 2 v 1 + 2h 2 2 + 1l - x2 2 v 2 (b) Tke the derivtive of t with respect to x. Set this derivtive equl to zero to show tht this time reches its minimum vlue when n 1 sin u 1 = n 2 sin u 2. This is Snell s lw nd corresponds to the ctul pth tken by the light. This is nother exmple of Fermt s principle of lest time (see Problem 33.56). 33.58.. Light is incident in ir t n ngle u (Fig. P33.58) on the upper surfce of trnsprent plte, the surfces of the plte being plne nd prllel to ech other. () Prove tht u = u. (b) Show tht this is true for ny number of different prllel pltes. (c) Prove tht the lterl displcement d of the emergent bem is given by the reltionship d = t sin1u - u b 2 cos u b where t is the thickness of the plte. (d) ry of light is incident t n ngle of 66.0 on one surfce of glss plte 2.40 cm thick with n index of refrction of 1.80. The medium on either side of the plte is ir. Find the lterl displcement between the incident nd emergent rys. Figure P33.58 t n n9 u u9 b ub Q Figure P33.56 n P d 1 x 2 y 1 u 1 u2 y 2 l 33.57.. CLC ry of light goes from point in medium in which the speed of light is v 1 to point in medium in which the speed is v 2 (Fig. P33.57). The ry strikes the interfce horizontl distnce x to the right of point. () Show tht the time required for the light to go from to is Figure P33.57 x 33.59.. ngle of Devition. The incident ngle u shown in Fig. P33.59 is chosen so tht the light psses symmetriclly through the prism, which hs refrctive index n nd pex ngle. () Show tht the ngle of devition d (the ngle between the initil nd finl directions of the ry) is given by sin + d = nsin 2 2 (When the light psses through symmetriclly, s shown, the ngle of devition is minimum.) (b) Use the result of prt () to find the ngle of devition for ry of light pssing symmetriclly through prism hving three equl ngles 1 = 60.0 2 nd n = 1.52. (c) certin glss hs refrctive index of 1.61 for red light (700 nm) nd 1.66 for violet light (400 nm). If both colors pss through symmetriclly, s described in prt (), nd if = 60.0, find the difference between the ngles of devition for the two colors. Figure P33.59 u9 h 1 u 1 v 1 v 2 u l u 2 h 2 2 2 u u u b u b n
1112 CHPTER 33 The Nture nd Propgtion of Light 33.60. thin bem of white light is directed t flt sheet of silicte Figure P33.60 20.0 flint glss t n ngle of 20.0 to the surfce of the sheet. Due to dispersion in the glss, the bem is spred glss Silicte flint out in spectrum s shown in Fig. P33.60. The refrctive index of silicte flint glss versus wvelength is Vcuum Vcuum grphed in Fig. 33.18. () The rys d nd b shown in Fig. P33.60 correspond to the extreme wvelengths b 1.0 mm shown in Fig. 33.18. Which corresponds to red nd which to violet? Explin your resoning. (b) For wht thickness d of the glss sheet will the spectrum be 1.0 mm wide, s shown (see Problem 33.58)? 33.61. bem of light trveling horizontlly is mde of n unpolrized component with intensity I 0 nd polrized component with intensity I p. The plne of polriztion of the polrized component is oriented t n ngle of u with respect to the verticl. The dt in the tble give the intensity mesured through polrizer with n orienttion of f with respect to the verticl. () Wht is the orienttion of the polrized component? (Tht is, wht is the ngle u? ) (b) Wht re the vlues of nd I p? f1 2 0 18.4 100 8.6 10 21.4 110 6.3 20 23.7 120 5.2 30 24.8 130 5.2 40 24.8 140 6.3 50 23.7 150 8.6 60 21.4 160 11.6 70 18.4 170 15.0 80 15.0 180 18.4 90 11.6 33.62.. IO Opticl ctivity of iologicl Molecules. Mny biologiclly importnt molecules re opticlly ctive. When linerly polrized light trverses solution of compounds contining these molecules, its plne of polriztion is rotted. Some compounds rotte the polriztion clockwise; others rotte the polriztion counterclockwise. The mount of rottion depends on the mount of mteril in the pth of the light. The following dt give the mount of rottion through two mino cids over pth length of 100 cm: Rottion ( ) d-glutmic Concentrtion l-leucine cid (g> 100 ml) -0.11-0.22-0.55-1.10-2.20-5.50-11.0 I totl 1W>m 2 2 f1 2 0.124 1.0 0.248 2.0 0.620 5.0 1.24 10.0 2.48 20.0 6.20 50.0 12.4 100.0 I totl 1W>m 2 2 From these dt, find the reltionship between the concentrtion C (in grms per 100 ml) nd the rottion of the polriztion (in degrees) of ech mino cid. (Hint: Grph the concentrtion s function of the rottion ngle for ech mino cid.) 33.63.. bem of unpolrized sunlight strikes the verticl plstic wll of wter tnk t n unknown ngle. Some of the light I 0 reflects from the wll nd enters Figure P33.63 the wter (Fig. P33.63). The refrctive index of the plstic Plstic wll wll is 1.61. If the light tht hs Incident sunlight been reflected from the wll into the wter is observed to ir be completely polrized, wht Wter ngle does this bem mke with the norml inside the wter? 33.64.. certin birefringent mteril hs indexes of refrction n 1 nd n 2 for the two perpendiculr components of linerly polrized light pssing through it. The corresponding wvelengths re l 1 = l 0 >n 1 nd l 0 >n 2, where l 0 is the wvelength in vcuum. () If the crystl is to function s qurter-wve plte, the number of wvelengths of ech 1 component within the mteril must differ by 4. Show tht the minimum thickness for qurter-wve plte is l 0 d = 41n 1 - n 2 2 (b) Find the minimum thickness of qurter-wve plte mde of siderite 1FeO # CO2 2 if the indexes of refrction re n 1 = 1.875 nd n 2 = 1.635 nd the wvelength in vcuum is l 0 = 589 nm. CHLLENGE PROLEMS 33.65... Consider two vibrtions of equl mplitude nd frequency but differing in phse, one long the x-xis, nd the other long the y-xis, x = sin1vt - 2 y = sin1vt - b2 These cn be written s follows: x = sin vtcos - cos vtsin y = sin vtcos b - cos vtsin b () Multiply Eq. (1) by sin b nd Eq. (2) by sin, nd then subtrct the resulting equtions. (b) Multiply Eq. (1) by cos b nd Eq. (2) by cos, nd then subtrct the resulting equtions. (c) Squre nd dd the results of prts () nd (b). (d) Derive the eqution x 2 + y 2-2xy cos d = 2 sin 2 d, where d = - b. (e) Use the bove result to justify ech of the digrms in Fig. P33.65. In the figure, the ngle given is the phse difference between two simple hrmonic motions of the sme frequency nd mplitude, one horizontl (long the x-xis) nd the other verticl (long the y-xis). The figure thus shows the resultnt motion from the superposition of the two perpendiculr hrmonic motions. Figure P33.65 33.66... CLC rinbow is produced by the reflection of sunlight by sphericl drops of wter in the ir. Figure P33.66 shows ry tht refrcts into drop t point, is reflected from the bck (1) (2) p p 3p 5p 3p 7p 0 p 2p 4 2 4 4 2 4
nswers 1113 surfce of the drop t point, nd refrcts bck into the ir t point refrctive index of the wter in the drop. (d) rinbow will form C. The ngles of incidence nd refrction, u nd u b, re shown t points nd C, nd the ngles of incidence nd reflection, u nd when the ngulr deflection is sttionry in the incident ngle u tht is, when d >du = 0. If this condition is stisfied, ll u re shown t point () Show tht u nd ub C = u = u u C = u r,. b, b, the rys with incident ngles close to u will be sent bck in the. (b) Show tht the ngle in rdins between the ry sme direction, producing bright zone in the sky. Let u 1 be the vlue before it enters the drop t nd fter it exits t C (the totl ngulr of u for which this occurs. Show tht cos 2 u 1 = 1 3 1n2-1). (Hint: deflection of the ry) is =2u - 4ub + p. (Hint: Find You my find the derivtive formul d1rcsin u1x22>dx = the ngulr deflections tht occur t,, nd C, nd dd them to 11 - u 2 2-1>2 1du>dx2 helpful.) (e) The index of refrction in wter get. ) (c) Use Snell s lw to write in terms of u nd n, the is 1.342 for violet light nd 1.330 for red light. Use the results of prts (c) nd (d) to find u 1 nd for violet nd red light. Do your Figure P33.66 results gree with the ngles shown in Fig. 33.20d? When you view the rinbow, which color, red or violet, is higher bove the u horizon? 33.67... CLC secondry rinbow is formed when the incident light undergoes two internl reflections in sphericl drop of u b wter s shown in Fig. 33.20e. (See Chllenge Problem 33.66.) () In terms of the incident ngle u nd the refrctive index n of the u u drop, wht is the ngulr deflection of the ry? Tht is, wht is ir Wter the ngle between the ry before it enters the drop nd fter it u r exits? (b) Wht is the incident ngle u 2 for which the derivtive of u C with respect to the incident ngle u is zero? (c) The indexes of refrction for red nd violet light in wter re given in prt (e) of u C b C Chllenge Problem 33.66. Use the results of prts () nd (b) to find u 2 nd for violet nd red light. Do your results gree with the ngles shown in Fig. 33.20e? When you view secondry rinbow, is red or violet higher bove the horizon? Explin. nswers Chpter Opening Question? This is the sme effect s shown in Fig. 33.31. The drfting tools re plced between two polrizing filters whose polrizing xes re perpendiculr. In plces where the cler plstic is under stress, the plstic becomes birefringent; tht is, light trvels through it t speed tht depends on its polriztion. The result is tht the light tht emerges from the plstic hs different polriztion thn the light tht enters. spot on the plstic ppers bright if the emerging light hs the sme polriztion s the second polrizing filter. The mount of birefringence depends on the wvelength of the light s well s the mount of stress on the plstic, so different colors re seen t different loctions on the plstic. Test Your Understnding Questions 33.1 nswer: (iii) The wves go frther in the y-direction in given mount of time thn in the other directions, so the wve fronts re elongted in the y-direction. 33.2 nswers: () (ii), (b) (iii) s shown in the figure, light rys coming from the fish bend wy from the norml when they pss from the wter 1n = 1.332 into the ir 1n = 1.002. s result, the You Wter ir ctul position of fish pprent position of fish fish ppers to be higher in the wter thn it ctully is. Hence you should im sper below the pprent position of the fish. If you use lser bem, you should im t the pprent position of the fish: The bem of lser light tkes the sme pth from you to the fish s ordinry light tkes from the fish to you (though in the opposite direction). 33.3 nswers: (i), (ii) Totl internl reflection cn occur only if two conditions re met: n b must be less thn n, nd the criticl ngle u crit (where sin u crit = n b >n 2 must be smller thn the ngle of incidence u. In the first two cses both conditions re met: The criticl ngles re (i) u crit = sin -1 11>1.332 = 48.8 nd (ii) u crit = sin -1 11.33>1.522 = 61.0, both of which re smller thn u = 70. In the third cse n b = 1.52 is greter thn n = 1.33, so totl internl reflection cnnot occur for ny incident ngle. 33.5 nswer: (ii) The sunlight reflected from the windows of the high-rise building is prtilly polrized in the verticl direction, since ech window lies in verticl plne. The Polroid filter in front of the lens is oriented with its polrizing xis perpendiculr to the dominnt direction of polriztion of the reflected light. 33.7 nswer: (ii) Huygens s principle pplies to wves of ll kinds, including sound wves. Hence this sitution is exctly like tht shown in Fig. 33.36, with mteril representing the wrm ir, mteril b representing the cold ir in which the wves trvel more slowly, nd the interfce between the mterils representing the wether front. North is towrd the top of the figure nd est is towrd the right, so Fig. 33.36 shows tht the rys (which indicte the direction of propgtion) deflect towrd the est. ridging Problem nswer: 1.93 * 10 8 ms >