Homework assignments are due at 11 pm (building close). Each problem part (a,b,c or other logical division) is worth 4 pts.

Transcription

1 Hmewrk assignmens are due a 11 pm (building lse). Eah prblem par (a,b, r her lgial divisin) is wrh 4 ps. Labs are shwn in red H1. P0.1,3,6,14,18,0 Hin: n 18 use eq If yu need mre examples wih mplex numbers, ggle "mplex numbers urial". Fr mre examples n ver alulus, ggle "url and divergene". H. P1. 5 and prblems x,y belw. Prb x. If yu wan reae an E-field disribuin and urrens in wires, find: a. ( x, yz,, ) ha is required E xyxˆ zyˆ 3 4 (7 )s frm a hanging harge b. B( xyz,,, ) ha mus be presen. Find sme B and J ha are nsisen wih his and shw ha E is a sluin he wave equain eqn [Ne: when yu find B and J nsisen wih all f Maxwell's equains, hen E will always be a sluin he full wave equain S in shwing i beys he wave equain, yu're shwing ha he wave equain saisfies Maxwell's equains in his pariular ase]. d. Sme f J will me frm hanging, and sme f J yu will have supply frm urren in neural wires. Find he erms in J(x,y,z) yu wuld have supply in wires (he divergene free erms). Prb y. The gray d represens an infinie sraigh wire f radius R ha has urren I ming u f I he page. We all his direin ẑ. Inside he wire here is unifrm urren densiy J = R (urren/area). a. Using a righ hand rule frm Phys 0, shw ha B is in he direin f ˆ (using ylindrial rdinaes r,, z). In her wrds B Br () ˆ, Find he srengh f he B-field fr r < R and r > R, in erms f I. See he hins n P1.3 b. In ylindrial rdinaes he url is wrien:

2 Using his and he B s yu fund in a), shw ha B J is rue. I. Nw we make J inrease wih ime, s ha B inreases wih ime: J () = R. is a nsan. B Frm E, shw ha here is an E-field indued alng he z axis, and find is magniude fr r<r. H3. P1.6, 9, L1.10 (8 ps), and Prbs 3x,3y belw. Prb3x. a) An elermagnei wave is raveling in he (, -1, 3) direin, and is E-field sillaes in he (1,, 0) direin. Ne ha neiher f hse vers has been nrmalized. The wave s eleri field ampliude is 5 V/m, is wavelengh is 13 m, and is speed is m/s. Wrie dwn a mplex expnenial plane wave whih desribes he eleri field f he wave. b) We an speify he wave s ha E ( 0 ) has any ampliude frm 0 he full ampliude A. Add a phase shif s ha a = 0 and r 0 he real par f E (whih we ake be he physial par) is ½ is maximum. E J P P Prb3y. a) Saring frm equ.13 E, using mplex plane waves and P E, learn he derivain (shwn in lass and ex) f hw k depends n he maerial: equ.17 k 1, and hene ha n 1. Praie unil yu an urn in a derivain yu have dne frm beginning end wihu any help frm ex, nes, r hers. b) Learn by hear he relains belw ha me frm he derivain abve and he definiin f k, and previus physis. When yu an wrie hem by memry, (use he blank se he righ praie as yu ver he answers), hen wrie a se urn in. When absrpin is negligible, n 1 n k n va v f n k f n va n k v

3 H4. P.3,5,7 and Prb 4.x. On.3 use Mahemaia and pl n( ), ( ). The ampliude f he harge displaemen will be muh less han an angsrm. Yu knw yu re ding i righ if ms f he values f n and a he end are beween 0.1 and. qn e 1 whih is fr a dieleri (insular), learn by me i qn e p hear he few ( r 3) seps and argumens we used in lass ge ni 1 1 me fr a meal wih small damping. Submi yur seps (wih heir argumens) when yu ve learned hem. Prb 4.x. a) Saring frm p b) Argue frm ni 1 ha ne f n, mus be zer belw, and he her mus be zer p abve i. Explain hw a purely real index vs a purely imaginary index rrespnd he ases f gd ransmissin vs pr ransmissin (in he ase f meals, he pr ransmissin shws up as reflein, n absrpin). H5. P.10, 3.,5 and Prb 5x,y belw. On 3. jus d i fr he firs w rws f equains, n all 4. Prb 5x. Exending P.10, assuming i s a green laser f = 500 nm whih fires 100 imes per send, find d) he number f phns in eah pulse e) he average pwer f he laser (averaged ver lng imes), f) he average number f phns per send i emis, g) he phn densiy (number/m 3 and number/angsrm 3 ) in he regin where he beam is fused 5 mm (use he E-field frm b). Bakgrund fr Prblem 5x bakgrund: The idea yu need is ha if here is sme energy in a beam, pulse r field, i an be expressed by a number f phns: energy N phns. S yu an relae his idea he pwer and inensiy f a laser beam. T relae i he field srengh iself, yu have wrie he inensiy r energy densiy in erms f bh he field and he phn energy. Fr exampleufield E, bu i N mus als be rue ha phns N u, where phns is he phn densiy (number/vlume). V V Prb 5y. Given fig 3.1 and having memrized he bundary ndiin ha E mpnens parallel he bundary mus be equal n bh sides, derive he fa f frequeny nservain, he law f reflein and Snell s law, i.e. eqns 3.4 hrugh 3.7. Inlude he neessary argumens. Turn i in afer yu an d i by hear. H6. L3.4 (8 ps), P3.1,13, and Prbs 6x,y. (On P3.1 and P6x, he physis we need is in he y and z dependene f eq 3.44). Prb 6x: exending P3.1 b) find he wavelengh (i.e. repea disane) f he evanesen wave ha prpagaes alng he surfae (see hin belw). ) n Mahemaia, pl he deay lengh (he lengh yu fund in he firs par f 3.1) vs angle in he riial regin. d) pl r and he phase f r vs angle in he riial regin fr bh s and p plarizain. Ne, n 3.1 and 13, fr bh jus use Fresnel s effiiens eq direly in Mahemaia, whih handles mplex numbers fine yu dn need eqn 3.49,50.

4 Hin n b): w ways d i draw he gemery f Fig 3.9 and d sme rig. Or ge i frm his idea: when we see a wave f he frm exp iau ( bvw), where u,v,w are rdinaes, we knw ha he far ha muliplies any rdinae mus equal, where d is he disane beween ress in he d direin f ha rdinae. Fr example a beause when u inreases by d d u (and he a u her variables dn hange), hen he wave has piked up a far f exp i 1, hene i s gne hrugh a full yle. Prb 6y: Turn hese in afer yu an d hem by hear a) Knwing ha he ransmied ray ges / a he riial angle, derive he riial iniden angle frm Snell s law. b) Fr iniden angles abve riial, derive he deay lengh (disane derease by far f e) f he evanesen wave wih his sraegy: If z is he nrmal he inerfae, shw hw Snell s law deermines an imaginary s ha is par f k z. Yur resuls shuld agree wih he z dependene in eq H7. P4.,3,4,5. On, add: b) give an explanain wih Phys 13 neps and skills (phase shifs and hiknesses) why he maxima and minima ur a he nes yu see. I's easies explain where R is large and small, whih rrespnds where T is small and large. ) Repea a and b fr an iniden angle frm air f 45 degrees. H8. P4. 6,8,10, L4.7. Simplify lab 4.y: a) Yu dn have pl anyhing, jus ake a few rais f inensiies nvine he grader ha T deays slwly fr d arund, hen expnenially fr d >> b) Suppse yu remve he nd prism, and uld auraely measure he deaying evanesen wave ampliude vs disane z frm he prism. Des he hery fr TIR give yu he same deay funin f he inensiy I(z) as frusraed TIR (T deays slwly fr z arund, hen expnenially fr z >>? Explain H9. P4.15, 17. On 17a,b: lis R fr all pssible hies f n. H10. 10y belw, phase mahing prblem R40 n pg, and 10x belw. On R40, he physis is ha all he n's grw wih frequeny, s an exrardinary index "ellipse" a ne frequeny migh inerse an rdinary index "sphere" a anher frequeny. Yu an d his wih plar pls in, bu maybe he easies way is pl vs nrmal pls f n-e(,), n-e(,), n-(), n-(). Of urse he n-'s dn' depend n, and will be sraigh lines. Lk fr inerseins f an n-e a ne frequeny wih an n- a anher frequeny. Prb 10y: Submi w lass review quesins nline fr he haper yu were assigned by . Please wrie n yur HW: 10y: I submied w quesins when yu are dne. TAs, I will send yu a py f he shee s yu an grade he mplein. Prb 10x. Ligh frm air eners a uniaxial rysal wih he pial axis alng he z axis shwn. The y direin is in he page. The pi axis is 45 degrees he rysal surfae. The iniden k-ver is in red: k k ˆ in xzˆ. The E-field (duble arrws) is in he plane f he 5

5 figure wih direin E xˆ zˆ in E 5. The pial prperies f he rysal are given by 3, 3, 8. x y z a) Find he numerial values fr he w nsans n, n e, and pl he funin ne( k OA) as vs. angle beween he OA and he unknwn k in he rysal. Als pl ne( ) where is measured frm he rysal nrmal. b) Find i in degrees frm he rysal nrmal. ) Find he angle frm Snell s law (measured frm he rysal nrmal). i.e. find where sin i (a nsan) and ne( )sin are equal. Wha is frm he OA? Chek: I g 5 deg frm OA axis. d) Knwing, wrie a uni ver fr k (in he x,z rdinae sysem). e) Frm k E P 0 find he rais f he mpnens f E in he rysal, and hene he direin f E in he rysal rdinae sysem. Find he angle beween E and k f) Knwing ha S and E are perpendiular, find he angle ha S makes wih he pi axis. Make a skeh shwing E, K and S in he rysal. Chek: I g S is 34 degrees frm he OA H11. P6.,4,5,6,8, and 11.x belw. Prb 11.x. 0.6 An iniial ligh sae is desribed by. x is he hriznal direin, y verial a) Desribe he iniial sae f he ligh in as muh deail as pssible. b) I srikes a quarer-waveplae wih is fas alng he y axis, and hen a linear plarizer ha ransmis a -45 degrees frm he +x axis. Wha frain f he iniial inensiy is ransmied? Use Jnes maries. ) Jus befre he final plarizer, desribe he sae f he ligh in as muh deail as pssible. H1. P7.1,3,4 and 1x,y belw. Prb 1x: Cmplee he plarizer aiviy. In lass yu will brrw hree linear plarizers. Yu an d his alne, r in pairs if yu d all he aiviies geher. Prb 1y: Suppse in sme regin he index is apprximaed by n n b 3 ( ). a) Find he phase veliy f a wave wih wavelengh in erms f hese symbls. b) Find he grup veliy f a wave wih average wavelengh. A uple f ways d his, bu yu migh use: k k Ans: vg 3 n b H13. P0.1,3,4 (Furier hery), P7.5

6 H14. P0.6,7,8,14x 14x. Cninues 8 abve. a) Find he Furier ransfrm f a f nsising f he sum f hree idenial Gaussian pial pulses f he frm =- 1. e /T sin, bu separaed by a ime 1. The hree Gaussians are enered a =0, = 1, and Use a nvluin herem. The hree-pulse funin f is a nvluin f e /T sin wih hree dela funins. Yu will use he resuls f 8 abve. Here is an easy hek: when 1 =0, he hree pulses are n p f eah her, s yu shuld ge 3x he answer in 8. b) Fr 1, T 8, 1 30 f. Cmmen n hw yur pl (and he pl in 8d) illusrae he neps we ve learned abu Furier ransfrms f pulses, and hw he w are differen, pl f. Pl he imaginary par f H15. P7.7,P8.,3,4,5, and Prb 15x: Ne 1 In 8. he hin says use. Every physiis mus knw hw derive his simple relainship by differeniaing k. Never use an yu see why i s wrng? ) Ne : n 8.4, fr he pling f I(), yu an se = 1, and use = 0.1. Ne 3: n 8.5, fr he pling yu an se = 1, and use = 0.1. Prb 15x: Cninue 8.4 add: a) Find frm eq Relae i yur pl and disuss wheher i beys he unerainy priniple disussed in lass. b) Frm eq. 8.16, find he fringe visibiliy a au and, and relae i yur pl. H15. P7.7,P8.,3,4,5, and Prb 15x: Ne 1 In 8. he hin says use. Every physiis mus knw hw derive his simple relainship by differeniaing k. Never use an yu see why i s wrng? ) Ne : n 8.4, fr he pling f I(), yu an se = 1, and use = 0.1. Ne 3: n 8.5, fr he pling yu an se = 1, and use = 0.1. Prb 15x: Cninue 8.4 add: a) Find frm eq Relae i yur pl and disuss wheher i beys he unerainy priniple disussed in lass. b) Frm eq. 8.16, find he fringe visibiliy a au and, and relae i yur pl. a H a, and 16x,y. Ne n 8.9a, and 16y if yu hange everyhing abu he sure angles ( max ) befre R yu ry inegrae, i shuld be easier. See he las slide f he leure. 16x: Cherene f sarligh The neares sar (her han ur sun) us is Prxima Cenauri a a disane f 30 rillin kilmeers, and i has an angular diameer f millinh f a degree r 7 milliarsends (1 milliarsend is 1 husandh f an arsend whih is ne sixieh f an arminue whih is ne sixieh f a degree hanks he Babylnians, wh lved muliples f 60). The nly reasn we knw is diameer is beause f inerfermery. a. If an asrnmer filers he whie sarligh hrugh a green filer, =500 nm, = 50 nm, wha are he apprximae empral herene lengh and herene ime f he ligh afer passing hrugh he filer? Use he unerainy priniple.

7 b. Using neps frm lass, skeh he pwer sperum I() fr his ligh, and label he apprximae widh in rad/se, as well as he psiin f he peak in rad/se.. Using neps frm lass, skeh he inerfergram I) fr delay fr his single sar. Label he apprximae widh f he wiggles in femsends (fs), as well as he apprximae herene ime. d. Wha is he apprximae spaial herene lengh f his ligh due he diameer f he sar? Use he simple esimae h. Use a single = 500 nm fr his analysis. This lengh is apprximaely he baseline f a sellar inerfermeer needed reslve he sar s diameer. 16y: In 8.9a) Insead f a line sure wih nsan inensiy, we nsider a -D dis wih nsan inensiy like he sun. a 4 y' We an sill d a ne-dimensinal prblem, if we use fr y', Iy' I 1, in her wrds, he inensiy is a prprinal he widh f he dis as we mve alng y, s he effeive 1-D inensiy is mre nenraed near y=0 a han fr a line sure. Fr y', I 0. i) Find (h) in erms f speial funins d i i Mahemaia. a ii) Using max, pl Re((h) vs h (in unis f ), and als pl Re((h)) fr he line sure in a). D his fr w R ases: max = 0.01 rad and rad. Whih has a lnger spaial herene lengh h, he line r dis sure? Why des ha make sense? H17. P9.3,4 and 17x: 17x: The ex s versin f he eiknal equain ( Rr nr sˆ r R by differeniaing, and ge ˆ ) is usually wrien in anher frm: we eliminae d ns n. This says ha he gradien f n deermines hw he direin ŝ f he ray ds hanges as yu mve a disane ds alng he pah. This frm an be used in a mpuer sluin rae any ray s pah. Le s resri n s i hanges in nly ne direin, fr example y. Then we an wrie nn' yˆ. If he direin f he ray in he x-y plane is given by, he direin is he uni ver sˆ s xˆ sin yˆ. d n ( y ) s n ( y ) ds a) shw ha ˆ ray pah. bemes 1 d n's ds. s is he variable ha ells yu where yu are alng he n d d d Hin:. Afer differeniaing, d bh sides wih sme uni ver ha will give yu a salar equain in ds ds d erms f d. 1 b) Argue (inluding a skeh f hw a uple f rays hange direin ) ha d n's ds means if he gradien n n desn hange sign, hen rays n aligned wih he gradien will evenually end up being aligned wih he gradien (y axis), if hey ravel lng enugh.

8 H18. P9.7,10,13,14 H19. P9. 16,18 H0. L9.14, 0.x,y,z belw 0.x Thin lens addiin frmula. Using eiher he ABCD al mehd r he prinipal planes mehd, jin w (r any number f) hin lens geher wihu any spae beween hem, and shw ha 1/f al = 1/f 1 + 1/f. Ne: his is why piians use "pial pwer" = 1/f fr a lens (uni 1/m alled a diper ), beause hey an add he "pwers" very easily frm he many es lenses ge he final presripin. 0.y Spherial wave addiin.. Our diffrain analysis mehds all rely n Huygen s priniple f adding E-fields frm spherial waves a every pin in an aperure. Here we will nsider jus hree equal srengh pin sures a y =a, 0 and a.. A sreen is plaed a z=d.. Eah sure emis a he same phase wih wavelengh +a (y,d) 0 z=d a) Salar addiin apprximain : Wrie he sum Eyd (, ) if eah sure emis a salar wave shape f f I( y, d) E * E if a 100, fr d a and d 15. a. ikr e A. Pl he R b) Fresnel (near field) regime. Yu need d>>as ha he angles he sreen are small. Keep dubling d and lk a he paerns. As lng as he shapes keep hanging (n jus expanding), his is he Fresnel regime. Pl w f hese ha lk prey differen and give he d's. ) Fraunhfer (far field) regime. Keep dubling d unil yu find a ase where he shape f he paern desn' hange (jus expands wih d, s i's he same angular frm). Pl w f hese ha lk similar fr very differen d's and give an apprximae d where yu see he Fraunhfer regime begin. Hw well des his mah he Fraunhfer bundary given in lass? 0.z a) Phased sli diffrain. Imagine a verial sli f widh a, illuminaed in a srange way: fr 0<x <a/, he aperure field is Ex' 1and fr -a/<x <0, E x' 1 ( u f phase, aused fr example by a piee f glass f he righ hikness ha vers half he sli). Find a funin ha is he shape f E x (yu an ignre leading nsans and phase fars). Pl he shape f inensiy I x vs. x in radians, fr he ase a = 500. E x 0 0 in ), and shw hw his urs frm yur funin in ), using using L Hspial s rule r limis fr small b. Jusify physially frm phases why -a H1. P10.5,6,7,11.5,7 Ne: n all f hese, sine hey are Fraunhfer diffrain, give he answers in angular frm I, r I x y,, and he pls as well (in radians), n as direed in he ex. On 11.7, ignre he senene ha alks abu a lens n needed if yu use he angular frm.

9 H. (Thurs) P11.3,.x, y x Fresnel Znes The semiirular phasr drawing belw shws E a he ener f a sreen as a irular aperure is pened mre and mre, unil he firs zne is pen. Finish he skeh fr he znes, 3, 4, 5, 6, using he "vibrain urve" a he righ as a guide fr he spiraling ha is due inreasing R and he bliquiy far. Will 6 znes filling he aperure give yu a brigh r dark sp? Wha is he inensiy a he sreen ener mpared he iniden inensiy (use he lengh yu ge frm he a) Explain why he disane frm saring pin he ener f he spiral is he srengh f he iniden E-field, E 0, ha yu ge if yu remve all aperures. b) Using measuremens frm he vibrain urve diagram, find he lengh f hree phasrs ha represen ligh field E a he sreen ener frm znes 1,3,5 respeively, in erms f E 0. If yu made a zne plae ha blked znes,4, 6 as well as all znes n>6, wha wuld he apprximae E field a he ener f he sreen be in erms f E? Wha wuld he apprximae inensiy be f he ligh a he sreen ener mpared he inensiy f he laser beam wih n aperure (his zne plae is nw aing as a devie fus ligh). ) Repea he previus quesin wih a plae made blk znes 1,3,5 and n>6, bu allw ransmissin hrugh znes,4,6. d) A irular aperure has a diameer f 40. Hw far away n he axis (in unis f ) des yur sreen need be have he Fresnel znes 1 6 fill he aperure? Ne: in he Pyhagrean riangle yu se up, ne side wuld be he radius, n he diameer. Wha is he widh f he 6h zne (x 6 - x 5 )? y. When he sreen disane z is large enugh (r he aperure small enugh), nly ne Fresnel zne fills a irular aperure f diameer D. Find ha z. I shuld mah he apprximae z we g in lass fr he bundary beween Fresnel and Fraunhfer diffrain. Fr bigger z s, here s n way make he ener f he sreen dark. Bundaries f Fresnel znes: H3. 3x, 3y

10 3.x: A He-Ne laser beam =63 nm has a beam wais f 1 mm a he ener f a aviy. a) A wha disane (frm he ener) will he beam radius be mm, and wha is he radius f urvaure f he wavefrns here? b) Wha is he divergene angle f he laser? ) Chse a lens (f) and is psiin frm he ener s ha we will have a new beam wais f 0.1 mm a he fus. There are many pssible answers, bu shw why yur hies resul in his beam wais. Fr his prblem, assume ha he lens diameer is always larger han he beam diameer, s ha he lens diameer is n limiing he ligh beam ha yu fus. 3y. A He-Ne laser beam =63 nm laser aviy has mirrrs wih radii 1m and meers a) Find he beamwais, Rayleigh range, and fus psiin (frm he mirrr wih radius 1m) when he lengh L f he aviy is.5m. b) Pl he beamwais vs L ver he enire sable regin f L s. H4. P13.1, 4.x,4.y,4.z belw. Las HW! 4.x Lngiudinal laser mdes. a. Derive he simple relain beween lngiudinal mde number and allwed frequeny m given half-wavelenghs fi beween he w mirrrs. Fr a HeNe laser, =63 nm, find he apprximae m-value when L=0.5m. b. If he gain bandwidh is gain =1.5 GHz FWHM (frm Dppler bradening f a h plasma) find hw many lngiudinal mdes are lasing. Find he herene lengh f he ligh ha is emied ( gain gives he frequeny spread.. If we use a Fabry-Per inerfermeer, we an sele jus ne f he lngiudinal mdes lase. The widh f his single-m line is deermined by he leakage ime f ligh frm he aviy, line (1 R), where R is he L refleane f he upu upler. Use R = 99.0%. Find he herene lengh f he ligh ha is emied. d. Esimae he phns/send ha pass a given pin in he exerir laser beam if i has a diameer f mm and a pwer 1 mw. Nw esimae he al number f phns inside he laser (rughly nsan). Yu an ge his knwing ha 1% f he phns ge hrugh he upu upler eah ime hey srike i, and als deduing hw many imes a send eah phn inside srikes he upu upler. 4y. Vauum mdes and blakbdy radiain a. Cmpare w frequeny regins near w differen visible phn energies: h a = ev and h b = 3 ev. Whih regin (ev r 3 ev) has he ms E/M wave mdes per herz per m 3, and by wha far? b. Wha are he average numbers f phns in eah mde and als he average energy/mde a ev and 3 ev if hey are in equilibrium wih he surfae f he sun a kt 1 ev?

11 . Using he abve infrmain, whih regin (ev r 3eV) has he greaes inensiy f blakbdy radiain frm he sun (prprinal average energy/herz), and by wha far? 4z. Calulae he apprximae seady-sae emperaure f he earh: a. Find he slar inensiy a he earh s surfae. Use he pwer frm P13.1, and pu i in a spherial area f radius earh-sun disane. b. Assume he earh absrbs all he energy frm he sun ha his i, using he rss-seinal area f he earh, and find his pwer absrbed (he earh absrbs as hugh i were a dis faing he sun).. In seady sae, he pwer absrbed frm he sun equals he pwer emied by he earh in spae. Find he emperaure a whih he earh emis his pwer. Fr emissin, i emis uward in all direins s use he surfae area f a sphere. Yu shuld ge smehing reasnable fr he earh s emperaure. Ne: Realiy is a lile mre mpliaed a full mdel wuld inlude reflein frm he earh s surfae, as well as reflein bak frm greenhuse gases, he emissiviy, and all f hese as a funin f frequeny r wavelengh.

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