Notes follow and parts taken from sources in Bibliography

Transcription

1 PHYS 381 Noes follow ad pars ake from sources Bblography Waves I geeral, waves ca geerally be hough of as dsurbaces of a medum whch carry eergy. lecromagec waves are oably dffere sce hey do o requre a medum o ravel. We ge a ravelg wave whe we have somehg ha s a fuco of he sum (or dfferece) of me mulpled by velocy ad poso. If we frs look a oe dmesoal waves, we see ha a wave ca be descrbed a ay fxed po me as jus a fuco of poso f(x). The shape of he wave wll mach hs fuco could be a se or e, or a expoeal, or ay oher shape (perodc or o). For he wave o ravel, however, we eed he dsurbace o be a fuco of x ± v where v s he wave s velocy. The fgure below shows wo plos of dffere expoeal fu. The frs s of he form e α ( x v ) whle he secod s decal excep for he sg of he v (velocy) erm; e α ( x v ) The frs wave moves from rgh o lef (owards egave values of x) whle he secod moves from lef o rgh (owards posve values of x). 1

2 PHYS 381

3 PHYS 381 The wave equao ha specfes how he wave ravels s a secod-order paral dffereal equao, whch ca be wre ( oe dmeso) he form show below; ψ x v 1 ψ You ca quckly verfy for yourself ha soluos o hs clude se ad e fu of (x ± v) ad her lear combaos. Because of he coeco bewee rgoomerc fu ad expoeals, we could also wre he soluo as ψ k ( x v ) ( x ) e, where k s a a. Oher useful as relaed o k clude he wavelegh of he dsurbace (λ) whch s equal o π/k ad ω, he agular frequecy of he wave, whch s π f π v / λ k v. Varous combaos of hese are used by dffere exbooks o descrbe wave moo. The argume of he expoeal mus be (ad s) dmesoless, whch you ca verfy for yourself. Whe expressed as a rgoomerc fuco, he argume s radas (whch we ca gore dmesoal aalyss). Ths fuco of x ± v s kow as he phase of he wave, usually wre as φ. The wave fuco above s herefore more smply wre as ψ φ ( x, ) e We ca also add a addoal offse o he wave f, for example, s phase s o equal o zero whe x ad are zero. Ths s somemes kow as a phase offse or al phase, ad your book uses ε o represe (alhough oher coveos exs, such as usg φ as he al phase). Ths would he gve us a wave fuco of he form ψ ( φ ε ) ( x, ) e For example, f we are usg he Se fuco o represe he wave, ad he moo sars wh a amplude A a he org (x ) a me, we wll eed a offse of π/ (.e., ε π/). ψ (,) A s ( k * ω * ε ) As ε 3

4 PHYS 381 Two mpora wave characerscs are phase velocy v p (he speed ha a po of a phase, such as oe of he wave s peaks, has) ad group velocy, whch we wll dscuss laer. Because he wave ravels hrough boh space ad me, we ca hk of eher a sapsho of he wave (sa of me) showg a varao phase wh poso, or we ca hk of a amplude meer fxed a some po space havg a oupu ha vares wh me as he phase of he wave chages. We ca fd he speed of a po of a phase usg x φ φ x ± ω k ± v p Noce ha he sg of he phase velocy chages depedg o wheher he ravelg wave ravels o he lef or rgh, as should. Superposo & Ierferece Because wo soluos o he wave equao ca be learly combed ad sll form a soluo o, we ca alk abou he eraco bewee elecromagec waves by smple addo. If we wa o kow he magude of he wave formed by he combao of mulple waves, we jus add he szes of he dvdual waves. Ths s kow as he prcple of lear superposo, ad leads o some eresg effecs. (Whe he lgh volved s exremely ese, or cera maerals, we ca have he produco of olear effecs, bu we ll gore hem for ow). The red, gree ad blue waves below (dffere waveleghs ad phases) combe o produce he black wave. We descrbe he way he waves wll combe by lookg a her waveleghs ad phases. Usually, we ll look a sources producg he same wavelegh of radao, so phase dffereces are really he oly mpora hgs. For waves whch are phase (red ad blue), he e effec s o produce a wave of larger amplude (black) (rucve erferece) 4

5 PHYS 381 If he waves are ou of phase (red ad blue), he e effec s o produce a fla le (desrucve erferece) To ge sable erferece bewee wo sources, hey mus be cohere (meag he phase dfferece bewee he wo sources does shf back ad forh, bu raher says a. A commo way o do hs s o spl he lgh from oe source o wo sources by meas of a couple of sls.) If he sources are cohere, he wave amplude a a sgle place depeds o he dfferece dsaces bewee he observao po ad he wo sources. If ha dfferece s a whole umber of waveleghs, he observao po wll receve a peak from oe source a he same me receves a (dffere) peak from he oher source (rucve erferece). If he dfferece s a odd umber of half-waveleghs, a peak from oe source wll arrve a he same me as a valley from he oher source, gvg us desrucve erferece. 5

6 PHYS 381 Phasors ad Complex Numbers Oscllag pheomea (such as harmoc oscllaors, crcus, ad waves) are ypcally descrbed usg eher rgoomerc fu (se ad e) or expoeals. Ay complex umber of he form x y ca be wre he form Ae where A s he amplude of he umber ad s he phase. Accordg o uler, we could also wre he phase par of hs as e s Noce ha we ca exrac eher he real or magary par of hs expresso o ge or s. Also, keep md ha some exbooks (ypcally hose for egeers) use j sead of o represe (-1). Oe oher way o wre he complex umber Ae s A. I akes a lle pracce (or a good calculaor) o work wh complex umbers, bu hey are very mpora ools all pars of physcs. Because a wave aleraes from s posve maxmum o s egave mmum ad back o he maxmum oce each cycle, we ca mage he wave s sze as beg lke he shadow of he had of a large clock ha s llumaed from he 1 o clock poso, as show below: As you ca see, he legh of he shadow wll chage perodcally wh he frequecy of he sweep. The shadow s legh correspods o he momeary wave sze, ad he legh of he clock had s he amplude of he oscllao. Beg comforable wh boh ways of wrg complex umbers s mpora, because f we re addg/subracg wo or more complex quaes, wll be much easer o pu hem x y form ad combe he real ad magary pars separaely. O he oher had, f we re ryg o mulply or dvde wo complex umbers, s much easer o wre hem he form Ae ad mulply/dvde he A s whle addg or subracg he phases. Plae Waves Oe of he mos useful coceps opcs (ad quaum mechacs) s ha of plae waves. If you coec all of he pos o a wave fro whch have he same phase, he surface formed s a plae wave. I ges s ame from he fac ha, suffcely far from a po source, a seco of he sphercal wave produced wll look lke a plae, jus as a large feld may appear o be 6

7 PHYS 381 perfecly fla o he curved surface of he arh. The propagao dreco of he wave s ormal o he wave fro, ad s usually referred o as he wave vecor k. A plae wave passg hrough he po (x, y, z) wll sasfy he equao below;. a z k y k x k z y x For a wave movg purely he z dreco, hs reduces o k z z a. The a a wll chage as we move alog he z axs gog hrough peaks ad valleys (.e., as he phase chages). We could wre he ravelg wave as ( ) ( ) z k y k x k z y x Ae z y x ω ± Ψ,,, These waves are useful because we ca ruc ay hree-dmesoal wave from he proper combao of plae waves. Maxwell s quaos The equaos whch ulmaely gover opcal pheomea (ad all elecromagec waves) are kow as Maxwell s quaos whch ca be wre eher egral or dffereal forms. The egral form of hese equaos free space (where here are o curres or charges) s A d A d B A ds B dl A ds dl B ε µ The dffereal form of hese equaos uder he same codos would be B B B ε µ Usg vecor calculus dees, we ca exrac he followg from he dffereal forms: ( ) ( ) B ε µ 7

8 PHYS 381 Ths gves us he wave equao for (ad a smlar oe for B): µ ε The wave equao s vald for each of he sx compoes of /B. The velocy of he wave descrbed by hem s 1 v ε µ c demosrag ha lgh s, fac, a elecromagec pheomeo. Because boh B ad are dvergece-free empy space, here ca be o compoe of eher feld he dreco of wave ravel. We ca go furher ad say ha, addo o beg perpedcular o he propagao dreco, ad B are perpedcular o each oher as well. The polarzao of a wave, defed as he dreco of s feld, ca herefore be compleely specfed by he magudes of he wo compoes perpedcular o s dreco of moo. The eergy flow of a elecromagec wave s relaed o he feld sreghs by S 1 µ B ε c B The vecor S s kow as he Poyg vecor, ad pos he dreco of he wave s ravel. For harmoc waves, ha wll gve S ε c B ( k r ω ) We are ypcally eresed somehg ha does vary wh me or dsplaceme que as rapdly as he magude of S. Averagg he magude of S over a whole umber of cycles (or over a me whch s large compared o he wave s perod) wll gve us he esy (or rradace) whch s I c ε B c ε c µ B If you remember, hs looks smlar o he expressos foud for eergy desy a capacor or a ducor excep for he facor of c (whch we expec whe gog from a eergy desy 8

9 PHYS 381 whch s J/m 3 o a esy, or power per area, whch s J/(m s). Tha eergy desy s, fac, he same as he radao pressure produced by a beam of elecromagec eergy. The rradace of lgh from a po source wll drop off uformly wh dsace. By ervao of eergy, s easy o show ha he esy falls as 1/r. The shor explaao s ha esy s power/area. Sce area creases as r, f empy space does o creae or absorb lgh, we are lef wh a esy droppg as 1/r. Blackbody Radao I your roducory course, you saw ha he hermal radao emed by a ho objec s largely depede of he composo of ha objec. The oly mpora hg s he emperaure. Because of hs, we ca draw blackbody curves whch are he parcular shapes of a graph of esy vs. wavelegh emed for a parcular emperaure. As you ca see below, he curves show have oceably dffere shapes a dffere emperaures: If we waed o do a lle b more mah, we could deerme he emperaure of a objec jus by lookg a s specrum. If we kow he esy of 7 m radao compared o he esy of 4 m radao, ha rao should gve us a uque blackbody curve correspodg o a uque emperaure (lke he ew fas-measurg ear hermomeers). I he early par of he h ceury, Max Plack suggesed ha he aoms whch were oscllag o produce hs radao could oly have cera parcular values of eergy (kd of lke he dfferece bewee a 3-way lgh bulb ad a bulb o a dmmer swch). Plack beleved ha he eergy of he oscllaor depeded o he frequecy of oscllao as 9

10 PHYS 381 h f where s a posve eger or zero ad f s he frequecy of oscllao. Possbly he mos eresg par of he equao s he h, ow kow as Plack s a. Is value s 6.66 x 1-34 J s, ad s credbly small sze gves us a h abou why s exsece ook so log o be realzed. Ths s smlar o he way ha we kow (ow) ha a glass of waer s really composed of may dvdual peces of waer (molecules). We do see waer as beg lumpy because he peces are so rdculously small o he scale of our seses. For mos of our purposes, we could der waer o be a couous subsace; we could ake a ler of waer, dvde o wo equal pars, he dvde he half-ler pars half aga, ec. I would ake some very specalzed ad expesve equpme for us o ge o he po where we separae wo waer molecules from oe aoher ad ca he go o furher dvdg he waer. Jus as he sze of he aom ells us abou he lumpess of maer, Plack s a ells us abou he lumpess of eergy. Phoos Wha we have doe so far s o der lgh as a classcal wave. We ca jus as easly der o be a parcle. I fac, as he esy of lgh drops, he parcle aure of becomes more obvous. We ca use some of he deas above o dscuss a phoo flux whch (for a moochromac beam) wll be P/h f where f s he frequecy of he lgh, P s he beam power Was, ad h s Plack s a. Shorly afer Plack made hs fdgs kow, se realzed ha f he radag oscllaors ca oly have cera dscree values of eergy, hey mus be radag dscree values of eergy (f you ca oly have $1 blls your pockes, you ca gve someoe a quarer!). se called hese dscree peces of lgh eergy phoos. Oe of he early expermes where lgh seemed o be behavg as a parcle (or phoo) s kow as he phooelecrc effec. The basc dea here s ha whe lgh shes o cera meals, he phoos h elecros he meal hard eough o ejec hem from he res of he meal erely. se sad ha hs happeed because he phoos were carryg a eergy equal o h f whch should look famlar. We ca calculae he eergy of a red-lgh phoo as a exercse. For red lgh, f s abou 4 x 1 14 Hz, so h f gves us.65 x 1-19 J. Obvously, he Joule s a coveely large u of eergy for hs kd of suao, so we somemes use he elecro Vol (ev). We ve see hs before whe alkg abou he amou a elecro s eergy chages whe movg hrough a poeal dfferece of oe vol, ad s 1.6 x 1-19 J. O hs scale, our red phoo has a eergy of abou 1.7 ev. Blue lgh, a abou 7.9 x 1 14 Hz, would have a eergy of aroud 3.3 ev. 1

11 PHYS 381 The bg dea here s ha here s a coeco bewee he frequecy of lgh ad s eergy. Before hs, people would have assumed ha you could use somehg lke a dmmer swch o ur he esy of a lgh dow as low as you d care o, whou lm. Rug a blue lgh bulb off of hs dmmer swch would allow you o produce, for example, a a dm sream of blue lgh a a rae of 1 ev per secod, for example. The coeco bewee eergy ad frequecy says you ca do hs f you wa blue lgh (7.9 x 1 14 Hz, ayway), he smalles pece you ca have s 3.7 ev. Ayhg smaller wo be blue lgh. You wo see a a sream of lgh hs case, bu sead you ll see a dscree lump of blue lgh every few secods, wh ohg bewee. Ths s very much lke he suao wh aoms: he smalles pece of Uraum you ca possbly have has a mass of abou 4 x 1-5 kg. No oe ca gve you x 1-5 kg of Uraum, because jus does exs. You ca have x 1-5 kg of Hydroge, because he basc peces (aoms) are smaller. Smlarly, you ca have 1 ev of frared radao or mcrowave radao, ec., bu you ca have a 1 ev vsble-lgh phoo, X-ray phoo, ec. Geg back o he phooelecrc effec, was observed ha cera colors of lgh (ear he blue ed of he specrum) could rgger hs elecro ejeco, bu oher colors (ear he red ed) could. The esy of he lgh (umber of phoos per secod) would allow red phoos o ejec elecros, o maer how brgh he lgh bulb. Chagg he esy of a blue lgh bulb would chage he curre (umber of elecros per secod), however. A wave heory of lgh could expla hs. A slowly vbrag wave should (over a loger me) be able o delver he eergy eeded o free a elecro jus as well as a more quckly vbrag wave could do ( a shorer me). Also, o me lag was ever observed bewee urg o he lgh ad cachg he frs ejeced elecros. If waves were shakg hem loose, should ake some me o ge he frs oes movg. The phoo heory s able o expla all of hs. Frs, whle free elecros a meal may o be boud o parcular aoms, hey are ceraly boud o he overall body of he meal. If hey were oally free, you could pck up a pece of seel ad shake elecros ou of! I akes a ozero amou of eergy o remove a elecro from a meal. The exac amou depeds o he meal, bu s called he work fuco. For poassum, s abou.3 ev, whle for plaum, s closer o 6.4 ev. Wha wll happe f a red lgh phoo hs a poassum aom? Sce 1.7 ev <.3 ev, wo be able o ejec. The aom volved wll eher gore he phoo alogeher or absorb ad very quckly (aosecods) re-radae. Icreasg he esy of he red lgh jus meas more aoms are dog he same hg. Wha f wo phoos h he aom a abou he same me? The you could have a ejeco, bu he problem s ha o lgh source from he early h ceury was capable of hs kd of esy. The re-radao happes so quckly ha oly powerful lasers ca drop eough phoos o he surface quckly eough. For our purposes, red lgh wo gve us ay phooelecros. Wha abou vole lgh? Sce 3.3 ev >.3 ev, elecros wll be ejeced. I fac, here wll be eergy lef over afer he ejeco. Tha exra eergy goes o he kec eergy of he elecro, so leaves wh a greaer speed. Wh greesh phoos, havg eerges jus slghly above.3 ev, we would ge phooelecros, bu hey d be movg very slowly sce here s very lle eergy remag afer freeg hem. The formula s 11

12 PHYS 381 h f K max W where W s he work fuco for he parcular meal, hf s he eergy of he comg phoo, ad K max s he maxmum kec eergy a elecro ca leave wh. More ese vole lgh gves more phooelecros ejeced per secod whch meas a hgher curre. Dgal cameras use hs effec o gve us pcures whou flm. I aoher popular applcao, we ca collec hese eergec elecros ad use hem o power a elecrc crcu (a baery, afer all, s jus a source of movg elecros) ad we ll have a solar cell. Radao Pressure Jus as lgh carres eergy, also carres momeum ad ca herefore exer a pressure (radao pressure) o a objec. The chage a objec s momeum whe absorbs U Joules of eergy from lgh s jus p U c absorpo If he lgh s refleced sead of beg absorbed, he rasfer of momeum s wce as grea (hs meas ha lgh works he same way as mechacal objecs worked las semeser; a halsoe boucg off of a car roof rasfers more momeum o he roof ha a radrop of smlar sze ha hs & dras off): p U c refleco Usg dmesoal aalyss, we ca see ha esy s power/area ad pressure s force/area. We ca he wre P I c I ( abs) or Prad ( refl) c rad Ths s he dea behd he solar sal. The sal would be a huge shee of Mylar (he shy plasc maeral some helum balloos are made of) whch would ercep lgh from he Su ad reflec back, gvg he sal (ad whaever payload was aached o) a kck away from he Su. Alhough he rae of accelerao would o be hgh, he wegh savgs volved o havg o carry fuel are asoudg. 1

13 PHYS 381 Radao Whe a charged parcle s acceleraed, elecromagec eergy s radaed from he form of phoos (whch may be vsble lgh, rado waves, ec.). As s show your book (Fgure 3.8), he -feld les from a charge are dreced radally from s poso a he rearded me r/c. I oher words, you do observe charges he way hey are ow, bu raher he way hey were a a me r/c he pas, where r s he dsace bewee you ad he prevous locao ad c s he speed of lgh. Also, f he charge s movg he x dreco relave o you, we kow from relavy ha x wll be he same boh your frame ad he charge s frame, bu y ad z wll crease by a facor of γ. I oher words, he feld ges larger he dre perpedcular o he charge s velocy. If a charge moves wh uform velocy, here ca be o radao from. We ca see hs also from specal relavy by rasformg o he charge s referece frame (where ceraly ca radae, sce s eher accelerag or eve movg hs frame). The Poyg vecor (descrbg he radao from he charge) s obvously zero he charge s res frame, sce he ere B feld s zero here. If he charge acceleraes, here wll be a kk he feld les bewee he old velocy ad he ew velocy. Ths kk s a rasverse compoe of, ad s he source of he radao. Ths wll oly fall off as 1/r compared o he 1/r rae of decrease of he radal elecrc feld from a po charge. A log dsaces, herefore, wll be he doma erm. The mos mpora kd of radao for our purposes s kow as elecrc dpole radao. Ths s produced whe a par of oppose charges (formg a dpole) chage her separao. If he dpole mome (descrbed by he produc of he charge ad he separao) oscllaes susodally, we ca wre he dpole mome as a fuco of me as p ( ) p ( ω ) where p s he dpole mome whe he charges are a maxmum separao. Alhough he ear feld zoe s more complcaed, he radao zoe s smpler. The elecrc feld s descrbed by p k 4π s ε ( kr ω ) Noce ha he feld s, fac, proporoal o 1/r, ad ha o he axs jog he dpoles (where ), he radaed feld drops o zero as well. r 13

14 PHYS 381 The rradace s herefore gve by I p ω 4 s ( ) 3 3π c ε r The esy s a srog fuco of frequecy, whch s he reaso why he sky s blue ad suses are red. Hgher frequeces lke blue are scaered very effecvely all aroud he sky, leavg oly he red ed of he specrum o make all he way hrough he hcker pah hrough he amosphere whe he Su s ear he horzo. Lgh Maer Whe a delecrc s prese, Maxwell s equaos are modfed so ha µ s replaced by µ ad ε s replaced by ε. Ths gves a dex of refraco of ε µ ε µ For may maerals, µ ~ µ so we ca use Maxwell s relao for whch s K where K s he delecrc a whch relaes he permvy of free space o he permvy of he medum. K s geeral a fuco of frequecy so he same s rue of he dex of refraco. Ths s he cause of dsperso, where dffere colors of lgh have dfferg speeds hrough dspersve meda, ad herefore refrac a dffere agles as hey pass hrough he medum. Ths bedg as a fuco of color s wha allows rabows ad prsms o spl whe lgh o s ue pars. The eraco of lgh wh maer ca resul refleco, rasmsso, or absorpo. If he phoo s eergy s below wha s ecessary o promoe a elecro o he ex hgher eergy sae, he absorpo ad re-radao of he phoo wll be elasc, meag here s o loss of phoo eergy. Because hs resuls a phoo comg from some dreco ad beg reradaed a esseally radom dreco, he process s kow as scaerg. Deser maerals more commoly absorb phoos ad he share he eergy hermally wh oher aoms collsos. Ths process s kow as eher dsspave absorpo or jus absorpo. The chace of a phoo beg absorbed (somemes called he cross seco for absorpo) goes up rapdly whe he phoo s eergy s close o he amou eeded o move a elecro o he ex hgher eergy level. Ths eergy wll also be re-radaed very quckly (ypcally 1-8 s or so) a gas or los o collsos a lqud or sold or very dese gas. As more aoms are packed 14

15 PHYS 381 ogeher, he chace for hs kd of absorpo creases. The presece of may aoms earby shfs her eergy levels ad provdes may more opporues for absorpo. The umber of avalable eergy levels creases proporoally o N, he umber of aoms prese. Also, whle he separao bewee eergy levels a solaed aom may be several ev, he separao bewee allowed levels a sold s more lke several ev / N, or more lke 1-3 ev. Ths level spacg s so rdculously small ha hese are rghly called eergy bads raher ha levels. Polarzao Whe a delecrc maeral s placed a elecrc feld, he arrageme of he posve ad egave charges makg up s alered. The posve charges wa o move he dreco of he feld, ad he egave charges wa o move upsream. Of course, he egave charges are ghly boud o he posve charges by exremely srog aomc elecrc felds, so hey ca acually separae geeral. Jus as roducory physcs, he addo of a ew force o a mass hagg from a sprg wll cause o move ul reaches a ew equlbrum po where he forces oce aga balace. Whe he dsplaced charges reach her ew posos, hey wll produce her ow feld due o her polarzao. Ths feld s represeed by P, ad s relaed o as show below ( ε ε ) P Srcly speakg, he permvy s acually a esor, o a scalar. Ths meas ha a real subsace may o respod he same way o decal elecrc felds from dffere dre. Sce he charge dsrbuos (egave ad posve) are o loger ceered o he same po, here s a dpole mome assocaed wh hs polarzao. I some cases, he dpole mome s permae, as may happe for dffere aoms formg a sgle molecule such as H O. Wheher a exeral feld s prese or o, he elecros prefer he oxyge aom o he hydroge aoms. There ca also be a duced dpole mome, such as would occur whe a prevously-eural molecule s subjeced o a elecrc feld. I may cases, he polarzao (or dpole mome per volume) s learly proporoal o he appled elecrc feld, so we ca wre P α Ths s rue lear meda. α s kow as he polarzably ad descrbes how srogly he subsace respods o appled felds. The va der Waals force ha you may have heard abou chemsry ca be udersood as he araco bewee flucuag dpoles. Sce we kow ha elecros are po parcles movg aroud her ucle (o crcles), a gve aom wll have a dpole mome a almos ay me. I wll be radomly oreed ad chagg rapdly, bu lke all elecrc dpoles, s elecrc feld wll fall off as he verse cube of he dsace o : 15

16 PHYS 381 dpole The presece of hs feld wll duce dpoles oher earby aoms/molecules, ad he sregh of hose dpoles s relaed as show above by P 1 3 r α The eraco bewee he dpole ad aoher dpole duced by s of he form p, so we ge somehg proporoal o, whch s proporoal o 1/r 6, he well-kow form of he va der Waals poeal. A susodal elecrc feld hg a delecrc causes he dpoles o ry o le up wh. Ths s he dea behd he mcrowave ove: waer s a polar molecule, so hose molecules wll ry o le up wh he prevalg elecrc feld dreco. Chagg back ad forh very rapdly causes hea o buld up he maeral as he waer molecules erac wh each oher ad oher (possbly o-polar) molecules. You ca see somehg smlar f you hold a mage ear a compass eedle. You ca move he mage back ad forh, ad he eedle wll ry o follow. Of course, sce he eedle has some roaoal era (hough may be small), wll oly be able o follow movemes up o a po; beyod ha, ca respod o he feld chages quckly eough. The same hg happes wh waer, where s ably o follow chages falls off a aroud 1 GHz (above he ~.4 GHz operag frequecy of mcrowave oves). Whle he era of a molecule lke waer may resrc o followg frequeces less ha 1 1 Hz, elecros (wh her much lower masses) are able o follow he feld chages up o he opcal rego. I s commo o represe he elecro hs suao as boud o s aom by a lear resorg force (lke a sprg). The force o due o he elecrc feld wll jus be F q. Sce he resorg force s opposely dreced, we ge he followg from Newo s secod law: q ( ω ) k x m x where he double-do oao represes he secod me dervave. We ca also wre k erms of he aural frequecy of he oscllaor ha we would fd for a ordary mass m suspeded from a sprg of sprg a k: ω k m so we ca wre 16

17 PHYS 381 ( ω ) m x m x q ω As mechacs, we ca assume (a leas up o some frequecy) he elecro wll oscllae a he same frequecy as he elecrc feld drvg so ha we predc Whe sered he equao of moo, we ge x ( ) x ( ω ) q m ( ) ( ) x ω ω If he feld s oscllag o faser ha he aural frequecy ω, he elecro wll keep up wh. Above ha frequecy, wll move 18 ou of phase wh he feld. The x ha we have foud correspods o he separao of he posve ad egave charges. We ca use hs o fd he dpole mome per volume (polarzao) as P q N x q N / m ω ω ( ) We ca combe hs wh he formulas a he sar of hs seco o ge N q 1 ( ω ) 1 m ε ω ω For ω > ω, he dex of refraco wll be less ha oe ad wll be greaer ha oe f ω < ω. The dex of refraco goes back ad forh from above o below oe a dffere frequeces, meag here s more ha jus a sgle aural frequecy. A beer model s oe lke N q j ( ω ) 1 m ε j ω j ω where he f j represe oscllaor sreghs ad he ω j are he resoa frequeces for each oscllaor. f 17

18 PHYS 381 Accordg o wha we have so far, we would ge a fe value for he dex of refraco f he cde radao mached he aural frequecy of he oscllaor, ad ha s obvously ophyscal. The reaso s ha he model we have for he elecro as a mass o a sprg s a lle oo smple. There s also a dampg force whch s proporoal o he elecro s velocy. Ths has a form smlar o wha we would use for a mass oscllag o a sprg a flud (whch could be ar). Pug -mγv he equao of moo ad realzg ha v dx/d would gve us N q ( ω j ) 1 m ε j ω j ω γ j ω sce each oscllaor could cocevably have a dffere dampg facor γ j. Icludg he flueces of earby aoms chages our resul o f 1 N q 3m ε j ω j ω f j γ j ω We ca he look a regos where absorpo s o mpora (ω s far eough from ω j ha ω j - ω s large compared o γω). If he added erm s relavely umpora, we ge (below he resoace frequecy) a dex of refraco ha creases wh frequecy, gvg ormal dsperso. As ω keeps creasg ul eveually ears a resoa frequecy, he dampg erm wll be he mpora par of he formula. I he eghborhood of a resoa frequecy, we have absorpo bads ad he dex of refraco decreases wh creasg frequecy. Ths s kow as aomalous dsperso. The lecromagec Specrum The dffere colors of vsble lgh ha we ca see represe dffere waveleghs (ad frequeces) of he chages he magec ad elecrc felds ha make up lgh. The formula coecg wavelegh, frequecy, ad wave speed s he same as was for soud, excep ha he wave speed s ow he speed of lgh sead of he speed of soud. Tha formula s f λ c where f s he frequecy of he lgh, λ s s wavelegh, ad c represes he wave s speed (we geerally use a c for lgh sead of v). For vsble lgh, he waveleghs rage from abou 38 m or so (vole) o abou 75 m (red). Tha meas he lgh s frequecy rages from 4 x 1 14 Hz (red) o abou 7.9 x 1 14 Hz (vole). 18

19 PHYS 381 The vsble par of he specrum s oly a y slce of wha s ou here. Movg from vsble lgh o lower frequecy elecromagec radao, we ge o frared radao frs. Ths s sesed (f s ese eough) as hea, ad s wha warms your had f you pu ear a lgh bulb. I s also wha allows polce deparmes & solders o see people oal darkess (gh vso goggles), ad s wha your remoe corol pus ou. How far does hs par of he specrum reach? There are o hard physcal boudares bewee ay of he regos of he specrum some are more defe ha ohers because of bology (we ca see red, we ca see frared) ad some by regulaos (he FCC or some eraoal sadards body deermes ha FM rado rus from 88 MHz o 18 MHz). As far as he physcs s cocered, s a smooh raso from low frequecy o hgh frequecy. As we go o frequeces he low ed of he frared, we eveually ge o he mcrowave rego (aroud Hz or so). The mcrowaves ha cook your food are hs rage, as are mos cordless phoes, cell phoes, ad wreless compuer eworkg equpme. Mcrowaves are somemes lumped o wha s called he rado rego of he specrum. Also par of ha s he FM rage meoed above. Broadcas TV (whch covers a much larger poro of he specrum ha he MHz or so alloed o FM rado) sars he FM rego ad reaches up o he mcrowave rego (you ca see hs for yourself f you re ever drvg hrough a area ha has a local TV sao broadcasg as chael 6 (o ecessarly cable chael 6) you ll be able o pck up he audo o he low ed of your FM rado). ve lower frequecy, we ge o AM rado, whch rus from abou 55 khz o abou 165 KHz or so. Below hs, here are specalzed rasmers whch ca peerae deeper o he ocea o commucae wh submares whou requrg hem o surface. Also a he low ed of he specrum s he 6 Hz radao produced by power les (ad jus abou everyhg plugged o hem). Wha s above he vole ed of he vsble specrum? Immedaely above s he ulravole rage. Ths s resposble for gvg people as (ad sk cacer), ad reaches up o abou 1 17 Hz. Above ha, we fd X-rays lke he kd used o look a your eeh ad boes. These ru up o abou 1 Hz. Beyod ha, he res of he way up, we jus alk abou gamma rays. These are some of he mos poeally dagerous producs from uclear wase. You mgh have oced ha seems ha he leas damagg kds of radao are clusered a oe ed of he specrum (rado) ad he mos damagg (UV, X-ray, ec.) are a he oher ed. Tha s o a cocdece we ll see laer ha he eergy of a dvdual pece of lgh (phoo) s proporoal o s frequecy. Hgh frequeces hgh eergy phoos whch ca damage us. I a vacuum, all of hese dsurbaces (collecvely called M radao or, somemes, jus lgh) ravel wh he same speed, whch s 99,79,458 m/s. Ths speed ca be measured wh moder equpme a ermedae-level physcs lab. 19

20 PHYS 381 Raylegh Scaerg Whe vsble lgh passes hrough a h gas lke he amosphere, s frequecy ω wll be lower ha ay of he resoace frequeces ω j of he molecules. Ths gves elasc scaerg, meag (jus as he dealzed cases roducory mechacs) he phoo loses o eergy he eraco. I he classcal lm, where he umber of phoos s asroomcal (o much of a approxmao for ypcal lgh eses), he molecules wll radae sphercally a he same frequecy as he comg lgh. Noe ha alhough he radao from each molecule s sphercal, hs s he large-phoo lm; alhough all dre may be equally probable, oly oe s acually realzed for a gve phoo s emsso. For ar a ormal amospherc pressure ad desy, s obvously rare for hs scaerg o happe. We kow hs because ar s esseally raspare; f he scaerg were sgfca, we could see very far sce he phoos from whaever you wa o look a wll have oly a small chace of makg udsurbed o your eye. Because (as we saw earler) he scaerg s a fuco of frequecy (ω 4, specfcally), dffere colors erac o dffere degrees wh he lgh. Blue s scaered more easly ha red, so whe he Su s whe lgh hs he amosphere, he blue poro s bouced all aroud (he fac ha he amosphere s desy flucuaes wh me ad poso s also mpora) whch makes he sky blue. Whe he sulgh goes hrough a hck layer of he amosphere, such as whe s o he horzo raher ha overhead, mos of he blue s scaered away, ad wha s lef s reddeed. Tha s why surses & suses (as well as moorses ad mooses) are usually so red. The low desy of ar he amosphere s mpora for hs scaerg all dre. Whe he molecules are far from (ad he ear/far scale s se by he wavelegh of he cde radao) oe aoher, as he upper amosphere, we ca ge scaerg ay dreco. If, o he oher had, molecules are ear oe aoher o he scale of a wavelegh (dese meda), he waves hey em ed o erfere wh oe aoher. I he dreco he lgh s ravelg, he erferece s more or less rucve. I dre a a agle o he propagao dreco, pah legh dffereces o a gve po from he varous scaers ca be very dffere (aga, hs meas dffereces large compared o λ). If he pahs are very dffere, he phases from he radao from each scaerer have a wde rage of values, meag erferece could be desrucve oe locao a oe sa ad rucve a earby locao or a a slghly laer me. If molecules are far from oe aoher, he scaerg perpedcular o he beam ca be large. I he dreco of propagao, erferece s more rucve deser maerals. There are smaller pah legh dffereces o a gve po from varous scaerers, ad ha correspods o smaller phase dffereces. Ierferece s very mpora, ad lgh eds o ravel hrough he subsace more or less s orgal dreco, wh lle scaerg oher dre. The more regular he spacg of molecules, he less scaerg away from he forward dreco s see. Sce molecules are much less ha oe wavelegh apar for our amosphere ad vsble lgh, s geeral very hard o see he beam of a flashlgh or laser uless s boucg off of some objec. Tha s why s much easer o see a laser beam f you sprkle chalk dus or somehg

21 PHYS 381 lke he ar ha s more lkely o scaer he lgh off o he sde (whch s where you ll be lookg from, uless he laser s amed a your eye) ha you ca see. For larger parcles, as he dsace across he parcle ges o be larger ha λ, molecules o each sde ow radae relavely far ou of phase wh oe aoher. The Raylegh-based wavelegh depedece sars o dsappear, ad wha remas s kow as Me scaerg. Me scaerg s wha makes may small (bu o molecule-szed) hgs appear whe, sce ulke Raylegh scaerg, s relavely depede of λ ad s also sroger he forward (cde lgh) dreco. Trasmsso Whe lgh s rasmed hrough a medum, we ve see ha he elecros he aoms of he medum wll oscllae wh he same frequecy as he cde lgh, ad wll herefore produce a secodary wave erferg wh he prmary wave. Ths combao s kow as he rasmed wave. As he wave moves hrough he medum, s phase chages. Our earler aalyss showed ha he elecros ca be expeced o oscllae exacly phase wh he drvg wave as s frequecy creases. A ω, he oscllaors wll be 9 behd he drvg wave. Addoally, he wave produced by he oscllaors (he secodary wave) wll be 9 behd hs, meag he secodary wave s 18 behd he prmary a ω. Desrucve erferece s characerzed by a 18 phase dfferece bewee wo waves, ad we kow ha ha leads o exco of he wave. I hs case, we call absorpo ad oce ha happes a ω as we predced he las seco. As he frequecy of he prmary wave rses above ω, he secodary wave wll be more ha 18 behd he prmary. Ths s dsgushable from a wave beg ahead of he prmary by less ha 18. 1

22 PHYS 381 As show he graphs above, we have a prmary wave (gree), a secodary wave (blue) whch s dsplaced o he rgh (meag as me moves from lef o rgh, he peaks he blue wave wll happe afer he peaks he gree wave), ad he sum of hose waves (red). Here, he secodary wave has oly 9% of he amplude of he prmary wave. The op lef graph shows he secodary ad prmary waves phase, so he resula rasmed wave s herefore also phase. O he op rgh, we have he secodary wave laggg behd he prmary wave by 9, meag he rasmed wave also lags he prmary wave (hough o by he same amou). The lower lef graph shows a phase dfferece bewee secodary ad prmary waves of 18, gvg us oly a very small rasmed wave (f he ampludes were equal, we would ge o resula wave a all). Fally, whe he blue wave lags by 7 as he lower rgh pael, we oce ha ow would be more aural o say ha leads he gree wave by 9. Aga, he rasmed wave leads by a lesser amou. The phase shf s equvale o a chage phase velocy, sce a rasmed wave laggg he prmary wave by mus arrve a a gve po afer he prmary wave would have f were a vacuum. If he rasmed wave lags he prmary wave, he dex of refraco s greaer ha oe ad v < c. If he rasmed wave s laggg so far ha s leadg he prmary wave, we ca say < 1 ad v < c. The phase shf s proporoal o he pah legh he objec. I geeral, he shape of a plo of versus frequecy wll look sor of lke a se wave (dsplaced alog he y axs). For frequeces well below ω, he elecros have lle rouble followg he prmary wave ad he phase of he secodary wave lags by he usual 9. Sce he amplude of he secodary wave s small hs low-frequecy case, he rasmed wave lags very lle, gvg a dex of refraco ear oe. As he frequecy creases, boh he phase lag ad he amplude of he secodary wave crease, furher slowg he rasmed wave ul he secodary wave s 18 ou of phase wh he prmary ad has s larges amplude, resulg maxmum absorpo of he wave ad mmum rasmsso. A sll hgher frequeces, he secodary wave leads he prmary, gvg he rasmed wave a speed v>c. A hgher frequeces, he secodary waves reur o very small ampludes sce he elecros ca follow he oscllaos whe ω >> ω. The rasmed wave s phase s he oly slghly ahead of he prmary wave, gvg ~1.

23 PHYS 381 Refleco of Lgh If we have a fla, shy objec lke a ordary mrror, lgh boucg off of behaves a very smple way. If we measure he agle of he comg lgh relave o he ormal of he mrror (a le perpedcular o he mrror), we ll see ha he agle of he comg (or cde) beam s he same as he agle of he exg (or refleced) beam. If you hk abou hs for a mue, lgh srkg a mrror behaves he same way as a baskeball srkg a hard wall, ad for he same reaso. Ths very smple law s called he law of refleco, ad we ca sae brefly as r The way a mrror reflecs lgh s kow as specular refleco. The surface s very fla everywhere, so eghborg rays are o scaered radom dre whe hey srke he surface. For hgs whch are shy, refleco s sll happeg, bu s called dffuse refleco. The surface s ueve o he small scale (remember ha lgh waves are very small) ad he refleced lgh wll sll obey he law of refleco, bu each dffere spo o he surface has a ormal pog a dffere dreco. Image gaherg may people o a gym formg hem o wo les. Gve each par of people a baskeball, ad have hem bouce-pass o oe aoher. I would be very easy o sychroze hs so ha everyoe was dog he same hg a he same me. Ths s lke specular refleco. Now ake hese wo les of people ou o a bumpy pach of grass ad ry. The baskeballs wll sll obey he same physcal laws, bu he ueve aure of he groud wll sed hem all dre (dffuse refleco). Sell s Law ad Refraco The chage speed whe lgh goes from oe medum o aoher produces some eresg effecs. Whe lgh goes from ar o waer (low o hgher ), wll be be a he erface 3

24 PHYS 381 (uless s comg alog he ormal o he erface). You ca udersad hs by realzg ha oe par of he wave slows dow before he oher, ad ha causes he ur. Thk of he lgh wave as a wheelchar gog from a rego of easy ravel (cocree drveway) o a rego where wll move more slowly (grass). If you move from he drveway o he grass a a agle, he wheel whch ouches he grass frs wll slow dow, whle he wheel sll o cocree maas s speed. The e effec s o ur he wheelchar, as show below (jus he wheels ad axle are show): Noce ha upo reachg he slower rego, he wheelchar urs closer o he ormal (a le perpedcular o he erface bewee he grass ad cocree he ormal would be up ad dow he page hs case). Lgh behaves he same way. We ll draw some lgh rays ad clude he ormal because we wa o be able o measure agles from he ormal. We have hree rays he pcure; a cde ray, a refleced ray, ad a refraced (or be) ray. Ths s he geeral suao whe lgh hs he erface bewee wo maerals; some s 4

25 PHYS 381 refleced ad some s rasmed hrough afer beg be. We already kow he relao bewee he cde ad refleced rays hey make equal agles wh he ormal, jus as hey dd whe we looked a mrrors. The relaoshp bewee he refraced ad cde rays s foud he form of Sell s Law, whch we ca sae smply as 1 S 1 S where 1 s he dex of refraco he frs maeral, s he dex of refraco of he secod maeral, ad 1 ad represe he agles (relave o he ormal) made by he lgh rays each medum. Usg hs, f lgh srkes he surface of a swmmg pool a a agle of 17 relave o he ormal, wha agle wll he rasmed (refraced) ray make wh he ormal? We jus eed o solve S 17 S for (we re akg for ar o be equal o 1 sce s so close o oe ayway). We already kow ha he dex of refraco for waer s 1.33, so we ge 1.7 for our agle. Noce ha we refer o maerals 1 ad, o o cde ad refraced waves ad maerals. Tha s because we use he same mah f you re a he boom of a pool wh a flashlgh shg up o he ar. Ths effec s resposble for he srage appearace of a spoo a glass of waer looks be, eve f s o. Why, he, ca we look ou wdows ad o see huge dsoros? No all of he comg lgh s perpedcular o he wdow paes, so should be refraced ad should dsor our vew. The ea hg s ha he bedg whch happes gog from ar o he glass of he wdow s reversed whe he lgh goes back o he ar from he glass. The lgh ray has bee dsplaced slghly, bu s sll gog he same dreco was. You mgh have oced ha he equaly bewee 1 ad ad he fac ha he Se fuco ca oly rage bewee 1 ad 1. I seems we ca wre a equao ha ca be sasfed. For example, wha f we have a laser poer uder he surface of a swmmg pool. If (waer) ad 1 (vacuum, or ar f we approxmae) ad our cde ray comes a a shallow agle of abou 7 o he ormal (almos skmmg he ar/waer erface), wha dreco wll he laser beam have whe leaves he pool? We ge: ( 7 ) 1 ϑ 1.33S * S We eed a agle whch has a se of abou 1.5! We ca do ha. Wha wll happe? All he lgh wll be refleced back o he pool. Ths s called oal eral refleco. We ca fgure ou he agle a whch hs frs happes, kow as he crcal agle, by lookg a he cde agle for whch he refraced agle s 9 (he maxmum). We ll fd Sϑ ( ) S 9 1 c or ϑ c S for 1 > 1 1 5

26 PHYS 381 We ca ake advaage of hs effec by makg a cable ou of plasc wh a hgh value of ad he wrappg wh somehg wh a lower value of. Ths wll make a lgh ppe or opcal fber. If lgh srkes he sdes a a agle greaer ha he crcal agle, ges bouced back o he cable. Usg hs les us gude lgh aroud corers, dow ppes, ec. Huyges s Prcple A useful way o descrbe he propagao of a lgh wave hrough a medum s Huyges s prcple, whch says ha every po o he wave fro (whch, as we have see, s he surface of pos of he same phase) acs as a radaor of sphercal waves. Sce here s o backward wave, hey are really hemsphercal. The combed surface of hese hemsphercal waves forms he propagag wave. Illusraos of hs prcple for refraco as well as dffraco (ake from Wkpeda) are show below. 6

27 PHYS 381 Noce ha he lef sde of he refraced wave he frs fgure llusraes he source of he bedg: he place where he wave frs h he secod maeral (lef sde) sars re-radag he hemsphercal waves before he re-radao o he rgh sde begs. Ths pus he lef sde ahead of he rgh sde, causg he ur. Ferma s Prcple Ferma s prcple s also kow as he prcple of leas me. I says ha, for a gve sarg po ad edg po, lgh wll choose he pah whch akes he leas amou of me. We ca quckly prove he law of refleco sarg wh hs posulae. h x f 7

28 PHYS 381 I he fgure above, he pah legh whe he lgh beam reflecs from a po x o he rgh of he eye wll be h x h ( f x) We ca se he dervave of hs fuco wh respec o x o zero o fd he po of mmum pah legh ad we ge h x x h f x ( f x) You ca quckly verfy ha x f / s a soluo o hs equao. Ths meas he agles of cdece ad refleco wll be equal. Your book does a smlar aalyss for refraco ad defes a opcal pah legh whch s he produc of he dex of refraco ad he dsace raveled. Ths explas he reaso lgh does o ake a sragh le bewee a po of emsso a medum wh dex of refraco ad he po of absorpo wh a medum wh a dffere dex of refraco r. A more accurae way o sae Ferma s prcple would be (as s foud your book o p. 19) a lgh ray gog from po S o po P mus raverse a opcal pah legh ha s saoary wh respec o varaos of ha pah. Ths slghly dffere saeme jus requres ha he frs dervave of he pah legh above be zero; he secod dervave could be posve, egave, or zero so ha mmal pah leghs are as vald as maxmal pah leghs or pos of fleco. Fresel quaos Vewg lgh waves as elecromagec felds (whch hey are), we ca derve some of our earler assumpos abou her behavor usg Maxwell s equaos: B B B µ ε If we wre he me ad space depedece of he ad B felds as ( ) ( k r ω ) ( ) ( k r ω r, e ad B r B e ), 8

29 PHYS 381 we ca see ha applyg he grade operaor s equvale o mulplyg by k ad ha akg he me dervave s he same as mulplyg by -ω. We could he rewre Maxwell s equaos as k k B B k ω B k ε µ ω The frs wo equaos ell us ha here s o compoe of eher or B he dreco of propagao (k). The hrd equao could be rewre as c B B k ω The fourh equao wll gve he same relao wh he usual reversal of sg for a cross produc. Your book coas may equaos whch ca be derved from he maeral above. We wll jus focus o wo he rasmace ad reflecace equaos ha wll gve us he probably for a lgh wave cde o some maeral o be refleced vs. rasmed. We have wo pars of hese equaos, for lgh havg a feld eher perpedcular o or parallel o he plae of cdece: T r R ad T r R where he subscrps ad refer o he cde ad rasmed waves. 9

30 PHYS 381 Complex Idex of Refraco If lgh s a medum wh a hgh dex of refraco ad hs a erface wh a maeral wh a lower dex of refraco a he crcal agle, he lgh wll be oally erally refleced back o he hgh- maeral. Ths happes because he wave s forbdde from propagag hrough he low- rego. The wave des off expoeally he low- rego, meag we could der o have a complex wave umber k. Ths wll gve us a decayg oscllao. Somehg smlar happes meals. We ca use Maxwell s equaos o wre (for a meal wh coducvy σ) x y z µ ε µ σ Ths looks very smlar o he wave equao for he elecrc feld free space. Oe oable dfferece s ha he permeably ad permvy are hose of he meal, o free space. The oher, larger dfferece s he presece of he frs me dervave of he elecrc feld. Ths s a dsspave erm, jus as he frco erm s proporoal o he frs me dervave of poso he case of a mechacal oscllaor. Smlarly, a elecrcal crcu, he ressace erm s assocaed wh he frs me dervave of he charge. We ca use some of he same mah we used wh he prevous wave equao f we replace he dex of refraco by a complex oe wre as ~ R I The soluo o he equao ca ow be wre as ω y / c e ( ω ( y c) ) I R / Because he wave esy s proporoal o, wll de off over a characersc legh 1/α c/( ω I ) as I α y ( y ) I e As meoed your book, hs legh ( he vsble rego) s much less ha oe wavelegh (.e., a few m or less). Why are meals so effce a exgushg cde lgh? 3

31 PHYS 381 Gog back o he mah we used descrbg he re-radao of lgh erms of a damped, drve harmoc oscllaor, we ca look a free elecros (whch are umerous a meal) as beg uboud, so here s o resorg force ad o aural frequecy ω. Ths gves her poso as a fuco of me as q ( ) ( ) x mω The egave sg meas ha he elecros are movg 18 ou of phase wh he cde wave, ad so he elecrc feld hey produce eds o cacel ou quckly. Noce ha hs happes a opcal frequeces for meals, whle oly would have happeed a a frequecy above he resoa frequecy a delecrc. We could add a erm o represe he effec of free elecros o he earler formula for he dex of refraco ad we would ge ( ) N q f e 1 ε m ω γ eω j j ω ω j ω f γ ω j where he subscrp e sads for coduco elecros. If he frequecy of he cde lgh s hgh ad boh he boud elecro corbuo ad he free elecro dampg erm γ e are small, we ge ( ω ) 1 ε N q mω The free elecros are o boud o ay parcular aom, bu hey are boud o he meal as a whole. They ca be dered o be a plasma. The resorg force exered wll cause oscllaos wh a characersc frequecy f he elecros are dsplaced. Ths frequecy s kow as he plasma frequecy. We ca use hs o rewre he expresso above as P ( ω ) 1 ω ω Above he plasma frequecy, wll be less ha 1 ad he meal wll become raspare. Colors The colors of everyday objecs ca ow be explaed by a combao of effecs. I some cases (sal, sugar, may fe powders, ec.) objecs are whe because of he frequecy-depedece of 31

32 PHYS 381 he Me scaerg from her ue parcles (whch are larger ha he waveleghs of lgh volved). The eergy levels of he parcles ha make up oher maerals merge o absorpo bads, allowg he subsace o reflec some colors of lgh bu absorb ohers. Ths leads o he dea of addve ad subracve prmary colors. You may kow ha a compuer or TV dsplay s made of a colleco of housads of y dos ha produce red, gree, or blue lgh. Whe varous eses of hese hree lghs are combed, hey add o produce almos he ere rage of colors vsble o he eye. I equal amous, hey wll appear o be whe (f brgh eough) or gray. Oher combaos clude equal amous of red & gree (producg yellow), equal amous of gree ad blue (cya) ad equal amous of red ad blue (magea). Ths s he addve color model. Whe you pr, however, your oupu reles o refleced lgh (you ca see he compuer scree a dark room, bu o he prou). Ths s subracve colorao, sce he maeral subracs some colors (by absorpo) from he cde whe lgh (all colors). The subracve prmares are cya, magea, ad yellow. If you oce, hese are he hree colors of pr carrdges used mos color prers (a sx-color scheme s also used some phoo-qualy prers for beer color reproduco). Combg magea ad yellow equal amous wll gve red, whle cya ad yellow wll produce gree ad cya & magea gve blue. Ths s easer o see hs graphc from hp:// 3

33 PHYS 381 Noce ha gree ad red add o gve yellow, as saed earler. How do cya ad magea combe o gve blue? Uder whe lgh, cya wll reflec boh blue ad gree. Magea wll reflec red ad blue. If he red par of he cde lgh s absorbed by he cya dye ad he gree par of he cde lgh s absorbed by he magea dye, he oly par boh wll allow o be refleced s blue. Ths s why combg he prmary addve colors gves whe lgh bu combg he hree prmary subracve colors gves black (all colors absorbed). arly color prers ook advaage of hs fac o pr everyhg wh oly cya, magea, ad yellow, sce hey could be combed o produce black ex. The mxg s usually mperfec, hough, ad pre-mxed black k a separae carrdge provdes beer ad cheaper resuls. Ths s also par of he reaso why graphc arss ad ohers who desg o compuer bu pr resuls o paper have o carefully calbrae her dsplay ad her oupu. The addve RGB colors o he scree may look very dffere from he eveual CMYK (K sads for key accordg o Wkpeda, ad he key color s black) oupu. Bblography Physcs 6 h do, Cuell & Johso Fudameals of Physcs 7 h do, Hallday, Resck, & Walker Opcs, 4 h do,. Hech hp:// hp:// hp://webphyscs.davdso.edu/apples/reard/reard_fl.hml hp://hyperphyscs.phy-asr.gsu.edu lecrc Crcus, 7 h edo, Nlsso & Redel 33