' 1.00, has the form of a rhomb with

Transcription

1 Problm I Rflcto ad rfracto of lght A A trstg prsm Th ma scto of a glass prsm stuatd ar ' has th form of a rhomb wth A th yllow bam of moochromatc lght propagatg towards th prsm paralll wth th dagoal AC of th rhomb s cdt o th fac AB (Fg ) Th bam s totally rflctd o th facs AD ad DC th mrgs through ts fac BC For th yllow radato th rfracto dx of th glass s 6 Fg A Drv th mathmatcal xprsso for th agl θ as a fucto of th rfracto dx of th prsm such that th total dvato of th bam that xts th prsm to b zro Udr th abov codto calculat th umrcal valu of θ dgrs ad muts f 6 5 p Th total dvato s zro oly f I R SI Ths mas that ' ad r' r (symmtrcal path sd of th rhombc prsm) Thrfor JJ ' AC SI I R(5 p) From th fgur blow ad I th : / r / so that r / 3 / (5 p) O th othr had / / (5 p) Pag of 9

2 Th rfracto law (Sll-Dscarts) s s( / / ) s( / 3 / ) or cos( / ) cos(3 / ) (5 p) Sc cos(3 / ) cos( / ) coscos( / ) ss( / ) [cos ( / ) ]cos( / ) s ( / ) cos( / ) 4cos ( / ) 3cos( / ) th rfracto law bcoms cos( / ) [4cos ( / ) 3] caot b accptd as uphyscal so th physcal soluto s Th soluto θ = π ( 8 ) 3 cos( / ) 95 (75p) 4 θ = 35 4 (5 p) Th prsm wth θ dtrmd abov ad th drcto of th cdt bam rma fxd but th atur of th lght radato chags bg formd ow of th yllow doublt of th mrcury Th two wavlgths hav th valus 579 m rspctvly 577 m Th rfracto dcs of th glass for ths wavlgths ar 6 rspctvly whr 4 3 Th lght rays that xt th prsm tr logtudally to a astroomcal tlscop adjustd for ft dstac A Drv th mathmatcal xprsso for th agular dstac btw th two mags s through th tlscop (frst as a fucto of θ ad th as a fucto of ad ) ad calculat ts umrcal valu 3 p W hav a fxd agl of cdc ( = costat) ad two agls of rfracto r ad r - dr corrspodg to ad + d From s = s r w ca wrt = d sr + cosr dr O th othr had from s = s r w ca wrt cos d = d s r + cos r dr Bcaus d (or as th txt of th qusto) s a vry small (ftsmal) quatty w ca approxmat = ad r = r but wth d' d (= ) ad dr' dr ( ) (5 p) W wll dmostrat latr (s Not blow) that dr = - dr Now lt us dtrm th agl btw th two mrgt rays of lght amly d' obtag s r cosr d d ( dr) wth dr tgr cos cos Th fal rsult s cos 3 sr d d (5 p) cos s W kow th mathmatcal xprsso of cos (θ/) ad aftr a lttl algbra ca b xprssd aothr form amly d 3 (5 p) Pag of 9

3 4 Numrcally: 5 rad (or 7 ) (5 p) Not: I th tragl JDJ w wrt that th sum of r agls s π (radas) amly (π/-α) + (π-θ) + (π/-β) = π α+β=π-θ = costat (for a gv rhombc prsm wth fxd θ) Thrfor dα = -dβ whr α = θ+ r (s th tragl AIJ) ad β = θ + r (s th tragl CI J ) Sc θ s fxd (our stuato) dα = dr ad dβ = dr so that dr =-dr (5 p) A3 y f ob If th focal dstac of th tlscop s objctv s f ob 4m drv th lar dstac y btw th two mags s th focal pla of th objctv ad calculat ts umrcal valu 5 p d 3 tgfob ( ) fob Numrcally: y = mm (5 p) B Rfracto but mostly rflcto B Total rflcto gomtrcal optcs Total rflcto occurs wh lght travls from a mdum wth rfractv dx to aothr o wth th rfractv dx at a cdc agl l whr l s th crtcal valu of th cdc agl calld lmt agl byod whch thr wll b o rfractd lght At total rflcto th tr rgy of th cdt lght bam gos to th rflctd bam B Drv th mathmatcal xprsso for th lmt agl 5 p Wh th cdc agl rachs th lmt agl l th rfracto agl wll b 9 I ths cas th scod Sll s rfracto law gvs sl so l arcs B Total rflcto lctromagtc optcs Elctromagtc optcs provs that bsds bg totally rflctd th cdt lght bam also ptrats th lss rfrgt mdum as a vasct wav Th charactrstcs of th rflctd ad th rfractd lght bams dpd o th agl of cdc as wll as o th ortato of th lctrc fld of lght wav (calld polarzato) For smplcty lt us cosdr that th lctrc fld tsty s prpdcular o th cdc pla as rprstd Fg Th dcs r ad t rfr to th cdt rflctd ad trasmttd proprts of lght wav whl k s th wav vctor gvg th lght propagato ortato Morovr xˆ ŷ ad ẑ ar th ut vctors of th chos Cartsa rfrc fram Pag 3 of 9

4 Physcal ot: Th prturbato producd by a pla moochromatc wav a pot spac at a crta momt of tm ca b wrtt as Er t E cos t k r or to smplfy calculatos th complx form tkr r t E whr ad th takg oly th ral part of th rsult Mathmatcal ot: For th complx umbr Fg z a b a b part It ca b wrtt as z a b z a b a b cos s modulus of th complx umbr z ad ta b / a B Evasct wav a s th ral part ad b s th magary z whr z s th B Kowg that th cdt wav s a pla ad moochromatc o charactrzd by th quato t k r r t E prov that th z mathmatcal xprsso for th vasct wav s t r t E t ad drv th xact xprsso for th attuato coffct α as a fucto of th cdc agl θ th lmt agl l ad th wavlgth λ of th cdt wav Also drv th xact xprsso for th phas φ of th vasct wav For th trasmttd wav whr Sc t t kt r r t E kt r yk t zk t t cos t ˆ s ˆ cos yy ˆ zz ˆ k y s z s s 5 p Pag 4 of 9

5 th cos s s s bcaus l Udr ths codtos th lctrc fld of th trasmttd wav ca b wrtt as t r t kt z Et s t kt y s Th + sg th frst xpotal has o physcal sgfcac bcaus thr s o wav at apprcabl dstacs from th trfac I cocluso th lctrc fld of th trasmttd (vasct) wav has th form z r t E whr t t k t s s th attuato coffct of th vasct wav ad t kt y s s th wav s phas Ths rsult shows that th wav travls alog th trfac (alog y drcto) ad that t s attuatd th z drcto (prpdcular o th trfac) Bcaus c v k k t k v v v c th B Ptrato dpth k s s s s l B Drv th mathmatcal xprsso of th dstac Δz from th trfac at whch th ampltud of th vasct wav s tms smallr tha at th trfac as a fucto of th cdt wavlgth λ ad calculat ts umrcal valu Th frst mdum s glass 6 th scod s ar ad th cdc agl of lght s 4 5 p Pag 5 of 9

6 z B3 Th phas spd of th vasct wav B3 Drv th mathmatcal xprsso for th rato v whr v s th phas v spd of th vasct wav ad v - th phas spd of th cdt wav ad comput ts umrcal valu for th cas of th cdc agl of lght of 4 75 p Cosdrg th phas of th vasct wav ts phas spd s t k t y s v v k s s k s t so th rqustd rato s v 6 v s B4 Th rgy trasfrrd from th cdt wav to th totally rflctd wav For ay valu of th cdc agl th rlatoshp btw th ampltud of fld of th rflctd wav ad that of th cdt wav was drvd by th Frch physcst August Frsl (788 89): E cos cos r E cos cos Physcal ot: If th prturbato producd by a wav a pot spac at a gv momt s xprssd usg complx umbrs th th wav tsty has th mathmatcal * * xprsso I ce E c E whr E a b s th complx cojugat of th complx umbr E a b Hr s th vacuum prmttvty ad c s th spd of lght vacuum B4 Prov that th totally rflctd wav has th sam tsty as th cdt wav 5 p Pag 6 of 9

7 Sc th cos s s s k cos cos k k Er E E cos cos k k For ay complx umbr of th sam form w ca wrt a b a b a a b b whr b ta a I cocluso E r E whr ta s s l cos cos k Udr ths codtos th tsty of th totally rflctd wav wll b Ir c Er c E I B5 Th Goos Häch ffct Wh a cdt wav bam wth a ft cross scto udrgos total rflcto at a trfac btw two mda th totally rflctd wav bam s latrally dsplacd o a dstac D (s Fg 3) that was masurd for th frst tm by Goos ad Häch 947 I Fg 3 th dsplacmt alog th surfac s s ad th Goos Häch shft s th latral shft D dcatd th dagram Ths s th Goos Häch ffct Th xplaato of ths latral shft s basd o th apparac of th vasct wav at th trfac ad ts propagato paralll to th trfac Pag 7 of 9

8 B5 Th latral shft Fg 3 B5 Drv th mathmatcal xprsso for th Goos Häch latral shft D admttg that th phas dffrc btw th totally rflctd wav ad th cdt o s zro at th trfac Cosqutly comput th umrcal valu of th dsplacmt s alog th trfac as a fucto of th wavlgth λ of th cdt lght f th frst mdum s glass 6 th scod s ar ad th cdc agl of lght s 4 p From Fg 3 t follows that D s cos If th cdc pot for th cdt wav o th trfac th phas of ts lctrc fld s th at th startg pot for th totally rflctd wav ts phas s r k y s Wth r ad kowg that ky k s w obta s s s l ta k s s cos Fally Pag 8 of 9

9 Th umrcal valu of s s s s l ta cos D ta s 55 B5 Tm dd for th total rflcto A altratv xplaato of th Goos Häch shft ca b gv trms of th tm dlay assocatd wth th scattrg of a radato puls at th trfac Th cdt radato puls s ot scattrd stataously by th surfac but rmrgs to mdum aftr a tm dlay τ durg whch th puls propagats paralll to th surfac ad s dsplacd by th dstac s B5 s v y v Drv th mathmatcal xprsso for th tm dlay τ ad calculat ts valu f th frst mdum s glass 6 th scod s ar th cdc agl of lght s 4 ad th moochromatc radato has th wavlgth 579 m Th lght spd vacuum s 8 c 3 m/s s s s cs c ta s s s cos l 6 4 s 75 p proposd by Prof Flora ULIU PhD Dpartmt of Physcs Uvrsty of Craova ROMANIA Assoc Prof Sbasta POPESCU PhD Faculty of Physcs Alxadru Ioa Cuza Uvrsty of Iaș ROMANIA Pag 9 of 9