42. (20 pts) Use Fermat s Principle to prove the law of reflection. 0 x c

Transcription

1 4. (0 ts) Use Femt s Piile t ve the lw eleti. A i b 0 x While the light uld tke y th t get m A t B, Femt s Piile sys it will tke the th lest time. We theee lulte the time th s uti the eleti it, d the tke deivtive t miimize tht time. Assume it A is t height bve the le, d it B is t height b bve the le. Assume tht the it A is bve the igi lg the le sue, tht the eleti us t x tht xis, d tht the it B is diste m it A tht xis. The time the th will be: t x b x Set the deivtive this time with eset t siti equl t ze: dt dx x x x x x b x x b si x si i i 0

2 43. (0 ts) Shw tht i Yug s duble slit exeimet, i the t slit is veed with dieleti ilm thikess t d idex eti, the etie iteeee tte is shited uwd by: shit 1 t whee is the seti betwee the slits. Assume smll gles, ssume the ilm is thi med t the slit t see diste, d d t side eletis i the dieleti ilm. SOLUTION t si( F smll gles we ssume tht the bttm bem tvels ext diste si(), d the t bem ges thugh th thikess t (with idex eti equl t the th 1 vuum i). Eh these will et the hse lg the bttm bem eltive t the t. The ttl hse lg is: But we e t sideig y eletis hee, s = 0. The ext th si() uses the lwe bem t lg i hse. Hweve, the t bem sees lge til th th the bttm bem by t t, usig the bttm bem t led i hse: si t t

3 Sie the idie tte ges s: I I s The iteeee mxim u t: m whee m is y itege. This etes the diti: si t t m mx si m t t mx si mx m t 1 F smll gles: mx m t 1 The mxim, d theee the etie tte, shit u (highe gles) due t the ilm.

4 44. (5 ts) A it sue light S is emittig sigle wvelegth d is situted smll diste d bve le mi. A see stds ml t the mi t diste L m S with L >> d. Fid the itesity light the see s uti height y bve the mi. see S d y mi The gemetil iteetti this blem is muh esie i yu elize tht the light whih elets the mi t the see etes vitul imge the it sue diste d belw the mi. I til xis is led lg the mi, this is like Yug s duble slit blem with seti d. Hweve, thee is th i the hse lg. The eleted ys will shw hse shit whih must be sideed: d si F smll gles, si() be eled by y/l, whee y is the height bve the mi: L 4dy L This hse diti be ut i the exessi idie:

5 I I s I I dy s L Nte tht the / hges the sies t sies: I I dy si L Gde: 6 its the th tibuti t the hse dieee, 6 ts the eleti tibuti, 6 ts etly swithig m t y, d 7 its luggig it exessi I etly (swith m s t si is t equied). 45. (15 ts). Desig thi dieleti ilm t miimize eleti white light t ml iidee glss ( = 1.515) i wte ( = 1.33). Detemie vlues the timum ilm thikess d idex eti. Bus +3:emmed dieleti this use. Sluti w = 1.33 t g = T ete eiiet tieleti tig m thi ilm, we eed (1) destutive iteeee betwee the ist d sed eletis d () eh eleti t hve the sme mlitude.

6 The sed diti be ud m the Fesel s Equti eleti t ml iidee: 1 1 Whee is the eltive idex / 1 bem mvig m medium 1 t medium. Sie the eleti m dieleti is t lge, we id by simly equtig eh itee: w g 1 1 w w 1 1 g g Simle exsi this exessi leds dietly t: g w 1.4 The ist diti equies: t The eet eletis the hse dieee be iged sie eh bem mkes e eleti t highe idex medium. Sie this tig is the visible, it shuld be desiged wvelegth i the middle the visible setum, suh s = 550 m. The equied ilm thikess is the: t 4 97 m Cylite!

7 46. (0 ts) This is imge s ilm tke by eleted white light. Exli:. why it is lul. b. why sme egis e blk (thee is stble s ilm thee, it is t utuig). ) It is lul beuse the thikess the s ilm vies ss the sue. Fm u disussi thi ilm iteeee, we kw tht the eleted idie is: I 0.16I 0 s S the eleti mxim will deed the hse t d, d d deeds the wvelegth:

8 b) Blk egis hve s ilm, but eleti. This is beuse the ilm is vey thi (t hes ze i the equti belw). I these egis, the til th dieee will be egligible, but the hse dieee will still be i, sie the extel eleti m the t the ilm udeges i hse shit, d the itel eleti m the bk the ilm des t. 1 t 47. (15 ts) A thi ilm MgF ( = 1.38) is desited t glss s tht its tieleti wvelegth is 580 m ude ml iidee. Wht wvelegth is miimlly eleted whe the light is iidet isted t 45 degees? Ude ml iidee the ist d sed eletis will el i the ttl th dieee is hl wvelegth: 1 F tw extel eletis, the et dieee due t eletis is ze. The th dieee is twie the thikess times the idex the ilm: t 0 1 S ml iidee t 580 m, the ilm thikess is: t 4 580m 105m I the light is iidet t 45 degees, tht ltes the th legth by t s ( ) (but the et hse due t eleti is still ze) t s 1 t 0 Use Sell s lw t get t = 30.8 degees. The wvelegth tht stisies this diti is: 4t s t 498m