Wave Optics 337. With phase reversals occurrig i the reflectios at both the air-oil bouary a the oil-water bouary, the coitio for costructive iterferece i the reflecte light is oil t where is ay iteger. Thus, the iiu ozero thickess of the oil which will strogly reflect the 53- light is ti oil ( 53 ) (.5), a () is the proper choice. 3. I a sigle slit iffractio patter, fore o a scree at istace L fro the slit, ark friges occur where yark L taθark si θark ( a) a is ay ozero iteger. Thus, the with of the slit, a, i the escribe situatio ust be a 7 L ( y ) 5.. 3 5. ark a the correct aswer is see to be choice. 4.. 4. Fro Malus s law, the itesity of the light trasitte through a polarizer havig its trasissio axis oriete at agle q to the plae of polarizatio of the iciet polarize light is I I cos θ. Therefore, the itesity trasitte through the first polarizer havig θ 45 45 is I I cos ( 45 ). 5I, a the itesity passig through the seco polarizer havig θ 9 45 45 is I (. 5I)cos ( 45 ). 5I. The fractio of the origial itesity akig it through both polarizers is the I I. 5, which is choice. 5. The spacig betwee successive bright friges i a ouble-slit iterferece patter is give by ybright ( L ) ( + ) ( L ) L, where is the slit separatio. As ecreases, the spacig betwee the bright friges will icrease a choice is the correct aswer. 6. As iscusse i questio 5 above, the frige spacig i a ouble slit iterferece patter is ybright L. Therefore, as the istace L betwee the scree a the plae of the slits is icrease, the spacig betwee the bright friges will icrease, a (a) is the correct choice. 7. As iscusse i questio 5 above, the frige spacig i a ouble slit iterferece patter is ybright L. Note that this spacig is irectly proportioal to the wavelegth of the light illuiatig the slits. Thus, with >, the spacig is greater i the seco experiet a choice (c) gives the correct aswer. 8. The bright colore patters are the result of iterferece betwee light reflecte fro the upper surface of the oil a light reflecte fro the lower surface of the oil fil. Thus, the best aswer is choice (e). 9. The reflecte light tes to be partially polarize, with the egree of polarizatio epeig o the agle of iciece o the reflectig surface. Oly if the agle of iciece equals the polarizig agle (or Brewster s agle) will the reflecte light be copletely polarize. The better aswer for this questio is choice ().. I a sigle slit iffractio patter, ark friges occur where yark L si θark ( a). The with of the cetral axiu is the istace betwee the locatios of the first friges o either sie of the ceter (i.e., betwee the ± ark friges), givig with of cetral axiu ( + ) a a a Thus, if the with of the slit, a, is cut i half, the with of the cetral axiu will ouble, akig () the correct choice.
338 Chapter 4 ANSWERS TO EVEN NUMBERED CONCEPTUAL QUESTIONS. The wavelegth of light is extreely sall i copariso to the iesios of your ha, so the iffractio of light arou obstacles the size of your ha is totally egligible. However, sou waves have wavelegths that are coparable to the iesios of the ha or eve larger. Therefore, sigificat iffractio of sou waves occurs arou ha-size obstacles. 4. The wavelegth of light travelig i water woul ecrease, sice the wavelegth of light i a eiu is give by, where is the wavelegth i vacuu a is the iex of refractio of the eiu. Sice the positios of the bright a ark friges are proportioal to the wavelegth, the frige separatios woul ecrease. 6. Every color prouces its ow iterferece patter, a we see the superipose. The cetral axiu is white. The first axiu is a full spectru with violet o the isie a re o the outsie. The seco axiu is also a full spectru, with re i it overlappig with violet i the thir axiu. At larger agles, the light soo starts ixig to white agai. 8. The ski o the tip of a figer has a series of closely space riges a swirls o it. Whe the figer touches a sooth surface, the oils fro the ski will be eposite o the surface i the patter of the closely space riges. The clear spaces betwee the lies of eposite oil ca serve as the slits i a crue iffractio gratig a prouce a colore spectru of the light passig through or reflectig fro the glass surface.. Suppose the iex of refractio of the coatig is itereiate betwee vacuu a the glass. Whe the coatig is very thi, light reflecte fro its top a botto surfaces will iterfere costructively, so you see the surface white a brighter. Oce the thickess reaches oe-quarter of the wavelegth of violet light i the coatig, estructive iterferece for violet light will ake the surface look re. The other colors i spectral orer (blue, gree, yellow, orage, a re) will iterfere estructively, akig the surface look re, violet, a the blue. As the coatig gets thicker, costructive iterferece is observe for violet light a the for other colors i spectral orer. Eve thicker coatigs give costructive a estructive iterferece for several visible wavelegths, so the reflecte light starts lookig white agai.. The reflecte light is partially polarize, with the copoet parallel to the reflectig surface beig the ost itese. Therefore, the polarizig aterial shoul have its trasissio axis oriete i the vertical irectio i orer to iiize the itesity of the reflecte light fro horizotal surfaces. 4. Due to gravity, the soap fil tes to sag i its holer, beig quite thi at the top a becoig thicker as oe oves towar the botto of the holig rig. Because light reflectig fro the frot surface of the fil experieces a phase reversal, a light reflectig fro the back surface of the fil oes ot (see Figure 4.7 i the textbook), the fil ust be a iiu of a half wavelegth thick before it ca prouce costructive iterferece i the reflecte light. Thus, the light ust be strikig the fil at soe istace fro the top of the rig before the thickess is sufficiet to prouce costructive iterferece for ay wavelegth i the visible portio of the spectru.
Wave Optics 339 PROBLEM SOLUTIONS 4. The locatio of the bright frige of orer (easure fro the positio of the cetral axiu) is ( y ) ( L ),, ±, ±.... Thus, the spacig betwee successive bright friges is bright, y y y L + bright bright + bright Thus, the wavelegth of the laser light ust be ( L ) 3 ybright (. 58 c )(. ) L 5. L 7 6. 3 63 4. (a) For a bright frige of orer, the path ifferece is δ, where,,,k. At the locatio of the thir orer bright frige, 3 a δ 3 3( 589 ). 77. 77 µ For a ark frige, the path ifferece is δ +, where,,,k. At the thir ark frige, a 5 3 δ + ( 589 ). 47. 47 µ 4.3 (a) The istace betwee the cetral axiu a the first orer bright frige is y L y y bright bright, or 9 L 546.. y 3. 5 3. 6. 6 The istace betwee the first a seco ark bas is y L y y ark ark. 6 as i (a) above. 4.4 The locatio of the ark frige of orer (easure fro the positio of the cetral axiu) is give by ( yark ) ( L )( + ), where, ±, ±,.... Thus, the spacig betwee the first a seco ark friges will be y ( y ) ( y ) ( L ) + L + L ark ark or 7 ( 5. 3 )(. ) 3 y 3. 53 3. 53 3. 3 4.5 (a) Fro siθ, the agle for the axiu for the sou waves is θ v si si sou s i f 354 s. 3 Hz 36. cotiue o ext page
34 Chapter 4 For 3.-c icrowaves, the require slit spacig is ( )( 3. ) siθ si 36. 5. 8 c (c) The wavelegth is si θ ; a if this is light, the frequecy is c c f siθ ( 3. 8 s) (. 6 ) si. 5. 8 4 36 Hz 4.6 (a) The locatio of the bright frige of orer (easure fro the positio of the cetral axiu) is ( y ) ( L ),, ±, ± K. bright, If ( ybright ) 3 ( ybright ) 5. 3 c, the 3( L ) 5. 3 c, a 5. 3 c ( 5. 3 ). 5 3L 3. 5 The separatio betwee ajacet bright friges is ybright L, or 9 589. 5 y bright 3. 5. 77. 77 c 7 5. 89 589 4.7 Note that, with the coitios give, the sall agle approxiatio oes ot work well. That is, si θ, ta θ, a θ are sigificatly ifferet. The approach to be use is outlie below. (a) At the axiu, δ siθ, y or siθ L + y 3 4 ( ) + ( 4 ) 55. 7 The ext iiu ecoutere is the iiu; a at that poit, or 5 δ siθ + θ si 5 si (. ) 5 55 7 3 7. 7 The, y ta. 7 7 54 so the car ust travel a aitioal 4. 4.8 The agular positio of the bright frige of orer is give by siθ. Thus, if the 7 bright frige is locate at θ whe 6. 6., the slit spacig is 7 ( ) 6. siθ si 6. 9.9 µ
Wave Optics 34 4.9 The locatio of the bright frige of orer (easure fro the positio of the cetral axiu) is ( ybright ) ( L ),, ±, ±, K. If the bright frige is locate at y 3. 4 whe. 5 a L 3. 3, the wavelegth of the light is ( ) 3 3 ( ybright ) 3. 4. 5 L 3. 3 7 5. 5 55 4. The agular eviatio fro the lie of the cetral axiu is give by y. 8 c θ ta ta. 737 L 4 c (a) The path ifferece is the δ si θ (. 5 ) si (. 737 ). 93 3.93 µ (c) 6 δ. 93 3. 9 643 Sice the path ifferece for this positio is a whole uber of wavelegths, the waves iterfere costructively a prouce a axiu at this spot. 4. The istace betwee the cetral axiu (positio of A) a the first iiu is L L y + Thus, L ( 3. )( 5 ) y (. ). 3 4. (a) The agular positio of the bright frige of orer is give by siθ, or θ si ( ). If 5, the first three bright friges are fou at θ si 3 5., θ si 4 6 5. 3, a θ 3 si 6 9 5. The agular positio of the ark frige of orer is give by si θ ( + ), or θ si [( + ) ],, ±, ±,K. If 5, the first three ark friges are fou at 3 5 si 5., θ si 3 4 5., a θ si 5 7 5. θ (c) The aswers are evely space because the agles are sall a θ si θ. At larger agles, the approxiatio breaks ow a the spacig is t so regular. 4.3 As show i the figure at the right, the path ifferece i the waves reachig the telescope is δ ( si α ). If the first iiu δ occurs whe θ 5., the α 8 θ + 9. + θ 4., a δ 5 35 siα si 4. Thus, h si 5. 48.
34 Chapter 4 4.4 (a) ta y 4. 5 L.8 3 θ 3. 5. 5 3 3 (c) θ ta. 5. 44 a siθ si (. 44 ). 5 The sie a the taget are very early the sae, but oly because the agle is sall. () Fro δ si θ for the orer bright frige, (e) θ si 5 4 si θ. 4 si. 44 6. 3 5 5 6. 3 si. 4 7 4. 7 (f) y L ta θ (. 8 ) ta (. 7 ). 6.6 c 5 5 7 63 4.5 The path ifferece i the two waves receive at the hoe is δ, where is the istace fro the hoe to the outai. Neglectig ay phase chage upo reflectio, the coitio for estructive iterferece is δ + with,,, δ 3 so i + 4 4 i. 75 4.6 (a) With phase reversal i the reflectio at the outer surface of the soap fil a o reversal o reflectio fro the ier surface, the coitio for costructive iterferece i the light reflecte fro the soap bubble is fil ( ) where,,,k. For the lowest orer reflectio ( ), a the wavelegth is water t ( + ) 4 (. 333 )( ) 6. 4 64 (c) To strogly reflect the sae wavelegth light, a thicker fil will ee to ake use of a higher orer reflectio, i.e., use a larger value of. The ext greater thickess of soap fil that ca strogly reflect 64 light correspos to, givig t ( + ) 3 64 3. 6 fil (.333) 36 a the thir such thickess (correspoig to ) is t ( + ) 5 64 6. fil (.333) 6
Wave Optics 343 4.7 Light reflectig fro the first (glass-ioie) iterface suffers a phase reversal, but light reflectig at the seco (ioie-glass) iterface oes ot have a phase reversal. Thus, the coitio for costructive iterferece i the reflecte light is t fil ( + ) with,,, K. The sallest fil thickess capable of strogly reflectig the iciet light is t i ( + ) 6. 4.756 fil 85. 4 4.8 (a) With air < oil < water, phase reversals are experiece by light reflectig at both surfaces of the oil fil, a upper air-oil iterface a a lower oil-water iterface. Uer these coitios, the requireet for costructive iterferece is t fil,,,k, a the requireet for estructive iterferece is t fil,,,k. To have the thiest fil that prouces siultaeous costructive iterferece of 64 a estructive iterferece of 5, it is ecessary that fil t ( 64 ) + 5 where both a are the sallest itegers for which this is true. It is fou that are the sallest iteger values that will satisfy this coitio, givig the iiu acceptable fil thickess as t i + fil fil ( 64 ). 5 5 ( ) (. 5) (. 5) 5 Fro the iscussio above, it is see that i orer to have a fil thickess that prouces siultaeous costructive iterferece of 64- light a estructive iterferece of 5- light, it is ecessary that 64 ( 64 ) + 5 or 5 + This gives. 5 or.5. + + 4.9 With coatig > air a coatig > les, light reflectig at the air-coatig bouary experieces a phase reversal, but light reflectig fro the coatig-les bouary oes ot. Therefore, the coitio for estructive iterferece i the two reflecte waves is coatig t where,,, For fiite wavelegths, the lowest allowe value of is. The, if t 77. 4 a coatig. 55, the wavelegth associate with this lowest orer estructive iterferece is coatig t. 55 77. 4 55
344 Chapter 4 4. Sice air < oil < water, light reflecte fro both top a botto surfaces of the oil fil experieces phase reversal, resultig i zero et phase ifferece ue to reflectios. Therefore, the coitio for costructive iterferece i reflecte light is t, or t where,,, fil fil Assuig that, the thickess of the oil slick is t 6.9 fil 33 4. There will be a phase reversal of the raar waves reflectig fro both surfaces of the polyer, givig zero et phase chage ue to reflectios. The requireet for estructive iterferece i the reflecte waves is the t +, or t ( + ) 4 fil where,,, If the fil is as thi as possible, the a the eee thickess is t 4 3. c 4.5 fil. 5 c This ati-reflectace coatig coul be easily coutere by chagig the wavelegth of the raar to.5 c ow creatig axiu reflectio! 4. (a) With air < water < oil, reflectios at the air-oil iterface experiece a phase reversal, while reflectios at the oil-water iterface experiece o phase reversal. Thus, with oe phase reversal at the surfaces, the coitio for costructive iterferece i the light reflecte by the fil is t fil ( + ), where is ay positive iteger. Thus, t ( 45)( 8 ) 8 fil.,,, 3, K + + + 3 The possible wavelegths are:. 6, 54, 35, K Of these, oly 54 (gree) is i the visible portio of the spectru. The wavelegths that will be ost strogly trasitte are those that suffer estructive iterferece i the reflecte light. With oe phase reversal at the surfaces, the coitio for estructive iterferece i the light reflecte by the fil is fil t, where is ay positive, ozero, iteger. The possible wavelegths are t fil. 45 8 8,, 3, K or 8, 46, 7, K 3 of which oly 46 (violet) is i the visible spectru.
Wave Optics 345 4.3 (a) For axiu trasissio, we wat estructive iterferece i the light reflecte fro the frot a back surfaces of the fil. If the surrouig glass has refractive iex greater tha.378, light reflecte fro the frot surface suffers o phase reversal, a light reflecte fro the back oes uergo phase reversal. This effect by itself woul prouce estructive iterferece, so we wat the istace ow a back to be oe whole wavelegth i the fil. Thus, we require that t fil or t 656. 3.378 fil 38 The filter will expa. As t icreases i t fil, so oes icrease. (c) Destructive iterferece for reflecte light happes also for i t fil, or t fil. 378 38 38 (ear ultraviolet) 4.4 Light reflectig fro the lower surface of the air layer experieces phase reversal, but light reflectig fro the upper surface of the layer oes ot. The requireet for a ark frige (estructive iterferece) is the t, where,,, air At the thickest part of the fil t. µ, the orer uber is 6 t. 546. 9 7. 3 Sice ust be a iteger, 7 is the orer of the last ark frige see. Coutig the orer alog the ege of cotact, a total of 8 ark friges will be see. 4.5 With a phase reversal upo reflectio fro the lower surface of the air layer a o phase chage for reflectio at the upper surface of the layer, the coitio for estructive iterferece is t, where,,, air Coutig the zeroth orer alog the ege of cotact, the orer uber of the thirtieth ark frige observe is 9. The thickess of the air layer at this poit is t r, where r is the raius of the wire. Thus, 9 t 9 9 6 r 4 4 4. 35 µ
346 Chapter 4 4.6 +, or Fro the geoetry show i the figure, R R t r t R R r. (. ). 3 3 3 9 8. 6 5 With a phase reversal upo reflectio at the lower surface of the air layer, but o reversal with reflectio fro the upper surface, the coitio for a bright frige is t + + +, where,,, air At the 5 th bright frige, 49, a the wavelegth is fou to be t. 6 + 49. 5 5 7 6. 5 6. 5 4.7 There is a phase reversal ue to reflectio at the botto of the air fil but ot at the top of the fil. The requireet for a ark frige is the t, where,,, air At the 9 th ark rig (i aitio to the ark ceter spot), the orer uber is 9, a the thickess of the fil is 9 5 t 9 6 4. 75 4. 75 µ 4.8 With a phase reversal ue to reflectio at each surface of the agesiu fluorie layer, there is zero et phase ifferece cause by reflectios. The coitio for estructive iterferece is the t + + fil, where,,, For iiu thickess,, a the thickess is t + 4 fil 9 ( 55 ) 9. 96 8 99. 6 4. 38
Wave Optics 347 4.9 There is a phase reversal upo reflectio at each surface of the fil a hece zero et phase ifferece ue to reflectios. The requireet for costructive iterferece i the reflecte light is the t fil, where,, 3, 5 With t. c, a fil. 38, the wavelegths itesifie i the reflecte light are t ( 38)( ) fil., with,, 3, Thus, 76, 38, 9. a oe of these wavelegths are i the visible spectru. 4.3 The trasitte light is brightest whe the reflecte light is a iiu (that is, the sae coitios that prouce estructive iterferece i the reflecte light will prouce costructive iterferece i the trasitte light). As light eters the air layer fro glass, ay light reflecte at this surface has zero phase chage. Light reflecte fro the other surface of the air layer (where light is goig fro air ito glass) oes have a phase reversal. Thus, the coitio for estructive iterferece i the light reflecte fro the air fil is t,,,,. Sice, the iiu ozero plate separatio satisfyig this coitio is fil. 58 t 9 4.3 I a sigle slit iffractio patter, with the slit havig with a, the ark frige of orer occurs at agle θ, where si θ ( a) a ±, ±, ± 3,K. The locatio, o a scree locate istace L fro the slit, of the ark frige of orer (easure fro y at the ceter of the cetral axiu) is ( y ) L taθ L siθ L a. ark (a) The cetral axiu extes fro the ark frige to the + ark frige, so the with of this cetral axiu is Cetral ax. with ( ark ) ( ark ) L y y a 7 5. 4. 5 3. L L a a 3 8. 8. The first orer bright frige extes fro the ark frige to the ark frige, or L ( ybright ) ( yark ) ( y ark ) L L a a a 7 ( 5. 4 )(. 5 ) 3 4. 5 4. 5 3. Note that the with of the first orer bright frige is exactly oe-half the with of the cetral axiu.
348 Chapter 4 4.3 (a) Dark bas occur where siθ a. At the first ark ba,, a the istace fro the ceter of the cetral axiu is y L L L taθ siθ a 6 (. 5 ). 4 9 3 3. 5. 3 The with of the cetral axiu is y (. 5 ) 4. 5 4.33 (a) Dark bas (iia) occur where si θ ( a ). For the first iiu, a the istace fro the ceter of the cetral axiu is y L L L a the eee istace to the scree is a L y. 85 3 3. 75 9 587.5. The with of the cetral axiu is y (.85 ). 7. 4.34 Note: The sall agle approxiatio oes ot work well i this situatio. Rather, you shoul procee as follows. At the first orer iiu, si θ a or 5. c θ si si 36 a. c 7. 98 The, y L ta θ ( 6. 5 ) ta 7. 98. 9 9. c 4.35 With the scree locatios of the ark frige of orer at ( yark ) L taθ L si θ ( L a) for ±, ±, ± 3, K the with of the cetral axiu is ycetral ( yark ) + ( yark ) ( L a), so a y cetral L axiu 3 (. 6 ).. 3 axiu 3 4.36 At the positios of the iia, siθ ( a) a y L taθ L siθ L a Thus, taθ p a L a 9. 5 68 3 y y 3. 3 7 4. 6 46 4. 7. 7
Wave Optics 349 4.37 The locatios of the ark friges (iia) ark the eges of the axia, a the withs of the axia equals the spacig betwee successive iia. At the locatios of the iia, si θ ( a) a y L taθ L siθ L a 9 5 (. ) 3.5 (. ) The, y (. ) a for successive iia,. Therefore, the with of each axia, other tha the cetral axiu, i this iterferece patter is with y (. ). 4.38 For iffractio by a gratig, the agle at which the axiu of orer occurs is give by siθ, where is the spacig betwee ajacet slits o the gratig. Thus, ( ) 63. 8 siθ si. 5 9 6. 8. 8 µ 4.39 The gratig spacig is ( 3 66 )c ( 3. 66 5 ) a siθ. (a) The wavelegth observe i the first-orer spectru is si θ, or θ si 5 3. 66 9 4 3.66 siθ This yiels: at., 479 ; at 3.7, 647 ; a at 4.8, 698 I the seco orer,. The seco orer iages for the above wavelegths will be fou at agles θ si si siθ. [ ] This yiels: for 479, θ. 5 ; for 647, θ 8. 3 ; a for 698, θ 3. 7
35 Chapter 4 4.4 (a) The logest wavelegth i the visible spectru is 7, a the gratig spacig is 6. 67 3. 67 6. Thus, ax 6 si 9.. 67 si 9.. 38 7 9 re so coplete orers will be observe. Fro si θ, the agular separatio of the re a violet eges i the first orer will be θ si re violet si si or θ. 9 7. 67 9 6 4. 67 si 9 6 c 4.4 The gratig spacig is. Fro siθ, the agular separatio 4 5 4. 5 5 betwee the give spectral lies will be or θ si θ si si re violet ( 656 ) 4. 5 9 5 si ( 434 ) 4. 5 9 5 The results obtaie are: for, θ 5. 9 ; for, θ 3. ; a for 3, θ 6. 5. Coplete orers for 4 are ot visible. 4.4 (a) c 4 6 If 6. 67 c 6. 67, the highest orer of 5 that ca 5 be observe will be ax 6 si 9 6. 67 3.3 or 3 orers 9 5 c 5 7 If 6. 67 c 6. 67, the 5 ax 7 si 9 6. 67. 33 or orer 9 5
Wave Optics 35 c 4 6 4.43 The gratig spacig is. c., a siθ gives the 5 agular positio of a seco orer spectral lie as siθ or θ si For the give wavelegths, the agular positios are θ si ( 9 ) 6 6. 9 6 48 37.6 a θ si. 8.7 If L is the istace fro the gratig to the scree, the istace o the scree fro the cetral axiu to a seco orer bright lie is y Ltaq. Therefore, for the two give wavelegths, the scree separatio is [ ] y L taθ taθ. ta 37. 6 ta 8. 7. 445 44. 5 c 4.44 With lies per cetieter, the gratig spacig is 4 6 c 5. c 5. The, fro siθ, the locatio of the first orer for the re light is θ si si 9 ( 64 ) 7 35 6 5.. 4.45 The spacig betwee ajacet slits o the gratig is c 5 3 slits 5 3 The axiu of orer is locate where siθ siθ y L + y ( )( 5 3 ), so. 488 (.7 ) + (. 488 ) 5. 4 7 54 4.46 Fro siθ, or siθ, we see that siθ siθ v r 8 violet siθ v3 3 violet re 4 3 siθ r 3 re Sice, for θ 9, θ icreases as siθ icreases, we have that θ < θ < θ < θ v v3 r r 3 so the seco a thir orer spectra overlap i the rage θv3 θ θ r.
35 Chapter 4 c 6 4.47 The gratig spacig is 3. 636. Fro siθ, or 75 75 θ si ( ), the agular positios of the re a violet eges of the seco-orer spectru are fou to be θ si r 9 re ( 7 ) si 65 6. 3. 636 a θ si v 9 violet ( 4 ) si. 7 6 3. 636 Note fro the sketch at the right that yr L taθr a yv L taθ v, so the with of the spectru o the scree is y L taθ taθ r v. Sice it is give that y.75 c, the istace fro the gratig to the scree ust be y. 75 c L taθ taθ ta (. 65 ) ta (. 7 ) r v Gratig Scree q q r v L y y v y r or L 9. 3 c c 4 4.48 (a). 38 c. 38 6 3 4. slits 9 6 589. siθ θ si si. 38 (c) y L ta θ (. ) ta ( 9. 65 ). 38 () θ si 9 589. 6 si. 38 y L ta θ (. ) ta ( 9. 69 ). 4 6 9. 69 9. 65 3 (e) y y y. 4. 38 (f) y y y L ta si. ta si. 537 3 ta si 9 ( 589. 6 ) 3 ta si 4. 9 ( 589. ) 3 4. The two aswers agree to oly oe sigificat figure. The calculatio is sesitive to rouig at itereiate steps.
Wave Optics 353. 4.49 The gratig spacig is. 5 3. 5 6. 4 Fro siθ, the agle of the seco-orer iffracte ray is θ si ( ). (a) Whe the gratig is surroue by air, the wavelegth is a θ si a 9 54 air si. 5 6 air 5. 6 air If the gratig is ierse i water, (c) the water water. 333, yielig θ si b 9 54 water si 9. 6 (. 5 )(. 333) Fro siθ, we have that siθ costat whe is kept costat. Therefore siθ siθ, or the agles of parts (a) a satisfy air a water Sell s law. c 4 6 4.5 The gratig spacig is 8. 33 c 8. 33. Usig siθ a the sall agle approxiatio, the istace fro the cetral axiu to the axiu of orer for wavelegth is y L taθ L si θ ( L ). Therefore, the spacig betwee successive axia is y y+ y L. b The loger wavelegth i the light is fou to be ( ) log 3 6 y 8. 44 8. 33 L. 5 469 Sice the thir orer axiu of the shorter wavelegth falls halfway betwee the cetral axiu a the first orer axiu of the loger wavelegth, we have 3 short L + logl or short ( 469 ) 78. 6 4.5 (a) Fro Brewster s law, the iex of refractio is taθ p ta ( 48. ). Fro Sell s law, siθ siθ, we obtai whe θ θ θ si siθ p. si 48. si 4.. p Note that whe θ θ p, θ 9. θ p as it shoul.
354 Chapter 4 4.5 (a) Fro Malus s law, the fractio of the iciet itesity of the upolarize light that is trasitte by the polarizer is I I I 5 ( cos ) (. ) θ av The fractio of this itesity iciet o the aalyzer that will be trasitte is I I cos ( 35. ) I (. 67) I (. 5)(. 67). 336I Thus, the fractio of the iciet upolarize light trasitte is I I. 336. The fractio of the origial iciet light absorbe by the aalyzer is I I. 5I. 336I I I. 64 4.53 The ore geeral expressio for Brewster s agle is (see Proble 57) taθ p (a) Whe. a. 5, θ p. 5 ta ta. 56. 7 Whe. 333 a. 5, θ p. 5 ta ta. 333 48. 8 4.54 The polarizig agle for light i air strikig a water surface is θ p ta. 333 ta 53.. This is the agle of iciece for the icoig sulight (that is, the agle betwee the iciet light a the oral to the surface). The altitue of the Su is the agle betwee the iciet light a the water surface. Thus, the altitue of the Su is α 9. θ 9. 53. 36. 9 p 4.55 (a) Brewster s agle (or the polarizig agle) is θ p quartz ta ta. ta 458. air 55. 6 Whe the agle of iciece is the polarizig agle, θ p, the agle of refractio of the trasitte light is θ 9. θ. Hece,θ p 9. 55. 6 34. 4. 4.56 The critical agle for total reflectio is θ c si. Thus, if θ c 34. 4 as light attepts to go fro sapphire ito air, the iex of refractio of sapphire is sapphire.. 77 siθ si 34. 4 c The, whe light is iciet o sapphire fro air, the Brewster agle is θ p ta. 77 ta. 6. 5
Wave Optics 355 4.57 Fro Sell s law, the agles of iciece a refractio are relate by siθ siθ. If the agle of iciece is the polarizig agle (that is, θ θ p ), the refracte ray is perpeicular to the reflecte ray (see Figure 4.8 i the textbook), a the agles of iciece a refractio are also relate by θ + p θ + 9 8, or θ 9 θ p Substitutio ito Sell s law the gives siθ p si 9 θ p cosθ por siθ p cosθ p taθ p 4.58 I I cos θ θ cos I I (a) (c) I θ cos I. I θ cos I 4. I θ cos I 6.. 4. 6. 45. 6. 65. 9 4.59 Fro Malus s law, the itesity of the light trasitte by the first polarizer is I I i cos θ. The plae of polarizatio of this light is parallel to the axis of the first plate a is iciet o the seco plate. Malus s law gives the itesity trasitte by the seco plate as I I cos ( θ θ ) I i cos θ cos ( θ θ ). This light is polarize parallel to the axis of the seco plate a is iciet upo the thir plate. A fial applicatio of Malus s law gives the trasitte itesity as ( ) ( ) I f I cos θ3 θ Ii cos θ cos θ θ cos θ3 θ With θ., θ 4., a θ 6., this result yiels 3 I f uits. uits cos. cos. cos. 6. 89 4.6 (a) Usig Malus s law, the itesity of the trasitte light is fou to be I I cos 45 I, or I I Fro Malus s law, I I cos θ. Thus, if I I 3 we obtai cos 3 θ or θ cos 3 54. 7 4.6 (a) If light has wavelegth i vacuu, its wavelegth i a eiu of refractive iex is. Thus, the wavelegths of the two copoets i the specie are a 546..3 546..333 43. 7 49. 7 cotiue o ext page
356 Chapter 4 The uber of cycles of vibratio each copoet copletes while passig through the specie are N 6 t.. 47 9 3.7 4 a N 6 t.. 44 9 49.7 Thus, whe they eerge, the two copoets are out of phase by N N. 4 cycles. Sice each cycle represets a phase agle of 36, they eerge with a phase ifferece of φ ( cycles)( cycle). 4 36 8. 6 4.6 I a sigle slit iffractio patter, the first ark frige occurs where si( θark ) a. If o iffractio iia are to be observe, it is ecessary that o real solutios exist for this equatio efiig the locatio of the first iiu. Thus, it is ecessary that si( θark ) a >., or a >. a a <. We the see that the axiu with the slit ca have before this coitio will fail a iffractio iia will start beig visible is a ax 63. 8. 4.63 The light has passe through a sigle slit sice the cetral axiu is twice the with of other axia (the space betwee the ceters of successive ark friges. I a ouble slit patter, the cetral axiu has the sae with as all other axia (copare Active Figures 4. a 4.6 i the textbook). Figure P4.63 I sigle slit iffractio the with of the cetral axiu o the scree is give by y L ta ( θ ) L si θ cetral axiu The with of the slit is the ark ark 9 L 63. 8. 6 a y. 3 7. 6 c cetral axiu L a L a c 4.. 4.64 (a) I a ouble slit iterferece patter, bright friges o the scree occur where ( ybright ) ( L ). Thus, if bright friges of the wavelegths 54 a 45 are to coicie, it is ecessary that 54 L 45 L or, iviig both sies by 9, 6 5, where both a are itegers. The coitio fou above ay be writte as ( 6 5). Trial a error reveals that the sallest ozero iteger value of that will yiel a iteger value for is 5, yielig 6. Thus, the first overlap of bright friges for the two give wavelegths occurs at the scree positio (easure fro the cetral axiu) 9 5 54. 4 y bright 3. 5 9 6 45. 4 3. 5. 5. 5 c
Wave Optics 357 4.65 Dark friges (estructive iterferece) occur where siθ + for,,,. Thus, if the seco ark frige occurs at θ 8. the slit spacig is i.. 3, 6. i + 9 3 546 siθ si. 3 v 4.66 The wavelegth is sou 34 s. 7. f Hz 4. 56. 56 Maxia occur where siθ, or θ si for,,,. Sice. 35, 486., which gives θ si (. ) 486. For,, a, this yiels axia at, 9., a 76.3. No solutios exist for 3 sice that woul iply siθ >. Miia occur where siθ ( + ) or θ si ( + ) With. 486, this becoes θ si +. 43. for,,, K. For a, we fi iia at 4. a 46.8. No solutios exist for sice that woul iply siθ >. 4.67 The source a its iage, locate. c below the irror, act as a pair of coheret sources. This situatio ay be treate as ouble-slit iterferece, with the slits separate by. c, if it is reebere that the light uergoes a phase reversal upo reflectio fro the irror. The existece of this phase chage causes the coitios for costructive a estructive iterferece to be reverse. Therefore, ark bas (estructive iterferece) occur where y ( L ) for,,, The ark ba occurs at y (that is, at irror level). The first ark ba above the irror correspos to a is locate at y 9 L 5. 3. 5. 5
358 Chapter 4 4.68 Assuig the glass plates have refractive iices greater tha that of both air a water, there will be a phase reversal at the reflectio fro the lower surface of the fil but o reversal fro reflectio at the top of the fil. Therefore, the coitio for a ark frige is t fil for,,, If the highest orer ark ba observe is 84 (a total of 85 ark bas coutig the orer at the ege of cotact), the axiu thickess of the wege is t ax ax 84 fil. 4 Whe the fil cosists of water, the highest orer ark frige appearig will be ax fil.333 tax 4 Coutig the orer, a total of 3 ark friges are ow observe. 4.69 I the figure at the right, observe that the path ifferece betwee the irect a the iirect paths is δ + x h With a phase reversal (equivalet to a half-wavelegth shift) occurrig o the reflectio at the grou, the coitio for costructive iterferece is δ ( + ), a the coitio for estructive iterferece is δ. I both cases, the possible values of the orer uber are,,,. (a) δ The wavelegths that will iterfere costructively are. The logest of these is for the case a has a value of + δ 4 h + ( ) + 6 6 4 5. 3 6. The wavelegths that will iterfere estructively are δ, a the largest fiite oe of these is for the case. That wavelegth is δ h + ( ) ( 5. ) + ( 3 ) 6 8.8 4.7 Fro Malus s law, the itesity of the light trasitte by the first polarizer is I I i cos θ. The plae of polarizatio of this light is parallel to the axis of the first plate a is iciet o the seco plate. Malus s law gives the itesity trasitte by the seco plate as I I cos ( θ θ ) I i cos θ cos ( θ θ ). This light is polarize parallel to the axis of the seco plate a is iciet upo the thir plate. A fial applicatio of Malus s law gives the trasitte itesity as cotiue o ext page ( ) ( ) I f I cos θ3 θ Ii cos θ cos θ θ cos θ3 θ
Wave Optics 359 (a) If θ 45, θ 9, a θ, the 3 ( ) I f Ii cos 45 cos 9 45 cos 9 If θ, θ 45, a θ 9, the 3 ( ) I f Ii cos cos 45 cos 9 45. 5 4.7 If the sigal fro the atea to the receiver statio is to be copletely polarize by reflectio fro the water, the agle of iciece where it strikes the water ust equal the polarizig agle fro Brewster s law. This is give by θ p water ta ta. 33 53. air Fro the triagle RST i the sketch, the horizotal istace fro the poit of refectio, T, to shore is give by x (. ) ta θ (. )(. ) 9 9 33 p a fro triagle ABT, the horizotal istace fro the atea to this poit is y θ 5. ta 5.. 33 6. 65 p The total horizotal istace fro ship to shore is the x + y + 6. 65 7. 4.7 There will be a phase reversal associate with the reflectio at oe surface of the fil but o reversal at the other surface of the fil. Therefore, the coitio for a ark frige (estructive iterferece) is t fil,,, + +, which reuces to r Rt t Fro the figure, ote that R r + R t r R Rt t. Sice t will be very sall i copariso to either r or R, we ay eglect the ter t, leavig r Rt. For a ark frige, t so the raii of the ark rigs will be fil r R fil R fil for,,,
36 Chapter 4 4.73 I the sigle slit iffractio patter, estructive iterferece (or iia) occur where siθ ( a ) for, ±, ±,. The scree locatios, easure fro the ceter of the cetral axiu, of these iia are at y L taθ L siθ L a If we assue the first-orer axiu is halfway betwee the first- a seco-orer iia, the its locatio is y + y + L a 3 L y a a the slit with is 9 3 L 3 5. 4 a 3 y 3. 4 3. 5. 35 4.74 As light eergig fro the glass reflects fro the top of the air layer, there is o phase reversal prouce. However, the light reflectig fro the e of the etal ro at the botto of the air layer oes experiece phase reversal. Thus, the coitio for costructive iterferece i the reflecte light is t ( + ) air. As the etal ro expas, the thickess of the air layer ecreases. The icrease i the legth of the ro is give by L t ( i + ) f + air air air The orer uber chages by oe each tie the fil chages fro bright to ark a back to bright. Thus, urig the expasio, the easure chage i the legth of the ro is air L ( ) ( ) Fro L Lα T 9 ( 5 ) 5 5., the coefficiet of liear expasio of the ro is L α L T 5 5. (. )( 5. C). 6 C