Module 5 - Thermal Radiation. A blackbody is an object that absorbs all radiation that is incident upon it.

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1 I. History of Blabody Radiatio A. What is a blabody? Modul 5 - Thrmal Radiatio A blabody is a obt that absorbs all radiatio that is iidt upo it. Wh radiatio falls upo a obt, som of th radiatio may b absorbd, rfltd, or trasmittd. Most obts that appar bla ar poor rfltors of optial radiatio but thy ar ot blabodis si thy still rflt som radiatio. A good approimatio of a blabody is a avity. I th ub blow, th radiatio trig th avity is otiually rfltd aroud th avity. Thus, th avity ats li a miow trap. It is asy for th light to tr but ot it. This is why a avity always loos bla at room tmpratur rgardlss of th olor of th isid of th avity. Frot Viw Sid Viw of Cavity B. Emissio Sptra Wh a body is at thrmal quilibrium, it must r-mit all th radiatio that it absorbs. Blabodis ar itrstig baus th olor of thir missio dpds oly o thir tmpratur ad ot upo thir shap or ompositio. Thus, thir missio sptrum a b usd as a pyromtr (tmpratur gaug). Th prossig of stl bam of grat importa durig th idustrial rvolutio of th lat 8's. Thus, blabody pyromtrs wr trmly usful tools for dtrmiig th tmpratur of stl ovs ad othr prossig quipmt.

2 Ergy Dsity Although light bulb filamts ad may othr rgular obts ar ot tru bla bodis, thir tmpratur a b approimatd from thir missio sptra by assumig that th obt is a blabody. Thus, blasmith's usd th olor of th hatd stl to dtrmi th prossig of stl wh maig buggy sprigs, hors shos t. C. Eprimtal Emissio Sptra Blabody Emissio Highr T Frquy W a s from th graph that th maimum missio frquy shifts to highr frquy as th tmpratur is raisd. W a also s that th total rgy mittd (ara udr urv) irass at irasig tmpratur. D. Stpha-Boltzma Equatio Th powr dsity mittd by a blabody is proportioal to th 4th powr of its tmpratur. P σt A 4 σ W K/m

3 E. Wi's Displamt aw Th pa of th blabody missio sptra is ivrsly proportioal to th tmpratur. ma.898 mm K T F. Ultra-violt Catastroph Th thory of blabody radiatio would appar to b a straight forward appliatio of lassial ltromagtism ad lassial physis. Although lassial physis produd a worabl thory at low frquy (log wavlgths), it faild at high frquis (short wavlgths). Si th thory prditd a ifiit amout of rgy radiatd at short wavlgths, it was alld th "ultra-violt atastroph." II. Dsity of Stats for A E&M Wav I A Blabody Radiator Si th missio sptrum of a blabody radiator is idpdt of th shap of th blabody, w ar fr to hoos th simplst shap possibl i drivig th dsity of stat futio. Thus, w hoos a ub with sids of lgth. = = Blabody (3-D Cub) -D E&M Wav Boudary (Aalogous to wavs o strig) A. Wav Equatio W start by writig th wav quatio for a wav i fr spa. W dvlopd this quatio usig Mawll's Equatios i Uivrsity Physis. E t E

4 whr î ĵ ˆ y z This quatio is a sparabl diffrtial quatio ad a b solvd by sparatio of variabls. This is a stadard math thiqu that you ithr may hav alrady studid or will lar latr i your ours wor. For ow, w ar oly ord with th fat that th ltri fild is a stadig wav that a b writt as th produt of thr spatial futios (X,Y,Z) ad a tim futio as show blow: E E B. Usig Boudary Coditios o X Zz Y y iω t From Uivrsity Physis, w ow that th ltri fild isid a prft odutor i ltrostati quilibrium is ZERO! Thus, w hav si boudary oditios that must b imposd upo our stadig wav solutio. Ths si oditios ar du to th si mtal odutors that form th fas of our ub. ) X() = ) X() = 3) Y() = 4) Y() = 5) Z() = 6) Z() = By looig at th figur of th stadig wav o a strig, you will oti that oditios,3, ad 5 ar satisfid by a si futio. Thus, w hav that X() = Si ( ) Y(y) = Si ( y y) Z(z) = Si (z z) W also s that oly rtai wav umbrs ('s) will satisfy th boudary oditios, 4, ad 6. Th argumt of th si futio must hag by a itgr umbr of radia as th spatial variabl hags by. This orrspods to th oditios that y y π π whr whr y is ithr,,,3,... is ithr,,,3,...

5 ,,,3,... ithr is whr π z z z Thus, w hav that th ltri fild i o dirtio a b writt as ω t z y o πz si π y si π si E E D. Wav Numbr Vtor - Th idividual wav umbrs ( y, z) foud i th prvious stio ar ompots of th thr dimsioal wav vtor as show blow: Th magitud of this vtor,, is alld th wav umbr ad is otd to th wav lgth,, by th quatio that w dvlopd i Uivrsity Physis: π z y W a ow ovrt this to a otio btw ad by z y π z y π y z

6 π π. Thus, oly rtai wavlgths ist i th avity as dtrmid by th magitud of th -vtor. You will study this matrial agai wh you dal with wav guids i th uior E&M lass. E. Dsity of Stats Futio W ow wish to dtrmi th umbr of availabl stats i th avity with wavlgths suh that is btw som valu to +d. A sphrial shll of radius ad thiss d has a volum of V 4π d Howvr, all ompots of must b positiv so w ar rstritd to oly /8th of th sphr's volum. π d V 4π d 8 W must also aout for th fat that a E&M wav a hav two diffrt polarizatio stats. Thus, th total umbr of stats with wav umbrs btw ad + d is G d π d

7 W ow ovrt this ito th umbr of stats i th avity btw th frquy ad + d usig th rlatioship. Substitutig this rlatioship ito our prvious rsults, w hav that d d 4π Gd d 8π d d 3 G 3 W ow divid by th volum of th ub to obtai th dsity of stat futio: gd 8π 3 d DENSITY OF STATES This is th umbr of allowd rgy stats pr uit volum of th avity that hav radiatio btw th frquy ad + d. Evryo usd this rsult i thir wor v Ma Pla. III. Calulatig Blabody Ergy Sptra To alulat th rgy pr volum pr frquy of th blabody, you multiply th avrag rgy mittd pr stat by th umbr of stats pr volum pr frquy. Thus, w hav that du εgv d

8 Thus, th problm with th lassial rsult had to ithr rsid i th alulatio of th dsity of stats usig Mawll's ltromagti thory or th avrag rgy alulatio usig lassial mhais (Mawll-Boltzma distributio / quipartitio thorm). IV. Classial Physis (Rayligh - Jas Thory) A. Th atoms of th blabody radiator ar osidrd to b lassial harmoi osillators whos rgy is giv by ε Notig that a lassial harmoi osillator has two dgrs of frdom ad usig th quipartitio thorm (from Mawll-Boltzma Statistis), w hav th avrag rgy of th osillator stats as ε Si th osillators ar i thrmal quilibrium with thir viromt, this is also th avrag rgy of th radiatio thy mit. B. Rayligh - Jas Radiatio Formula du ε g d 8π 3 Th formula a also b writt i trms of wavlgth by otig that d d.

9 Usig ths rlatios ad our Rayligh - Jas formula, w obtai for wavlgth du d 8π 3 8π 4 Thus, th lassial thory suggstd that rgy dsity i th avity is proportioal to th squar of th frquy or ivrsly to th fourth powr of th wavlgth. Thory fits th primtal data at low rgy but prdits ifiit rgy as or quivaltly. So far w hav alulatd th rgy i th avity volum ad ot th powr radiatd by th avity surfa. Som ttboos li Srway, Mosir, ad Moyr trat ths as th sam thig whih is fals. Thy ar howvr rlatd du to th fat that avity must b i thrmal quilibrium with its viromt or its tmpratur will hag. W will ow driv th powr mittd pr uit ara pr uit wavlgth by th avity as do i Rohlf s Modr Physis Alpha to Zta. Cosidr a diffrtial ara lmt of th avity walls blow: A A Th rgy pr uit wavlgth mittd is rlatd to th powr pr uit wavlgth pr uit ara by de = d d Δt ΔA = d C ΔA whr th is from th fat that radiatio may travl ithr dirtio to a wall. Howvr, th wav travls i all dirtios ad ot ust straight aross so w must osidr what happs for a agl thta. A θ A

10 This hags both th tim it for th wav to ma it aross ad th ara that th rgy is radiatd. Δt = Δt os θ, ΔA = ΔA os θ Thus, th amout of powr radiatd pr uit wavlgth pr uit ara at a agl thta a b foud by de d = d Δt ΔA θ os θ = d θ ΔA os θ d θ = ΔA de d os θ W ow avrag this ovr all possibl missio agls to fid th avrag powr radiatd pr uit wavlgth pr uit ara. d = ( ΔA de d ) π π os θ π dθ d = 4 ( ΔA de d ) Th quatity i th brat is rgy pr volum pr wavlgth of th avity that w drivd arlir. d = 4 du d Thus, w s that th powr dsity radiatd pr uit wavlgth ad th rgy dsity pr uit wavlgth of th avity ar rlatd by a proportioality ostat /4. iwis, w hav th sam proportioality ostat wh prssig th powr radiatd pr uit ara pr uit frquy. d = 4 du d

11 This ow givs us th tru Rayligh - Jas formulas: d = π 4 d = π C. Wi s Solutio ooig at th shap of th primtal data, Wilhlm Wi mad a hypothsis that th powr radiatd pr uit ara pr wavlgth for a avity was of th form d = a 5 whr a ad b ar ostats fittd to th data. From this rlatioship, h was abl to obtai both Wi s aw as wll as th 4 dpd at log wavlgth. Though th formula fit th iitial primtal data wll, latr primts i Grmay showd that th formula dviatd slightly from primtal valus at vry larg wavlgths. D. Ma Pla's Solutio Ma Pla obtaid th primtal data i Grmay ad immdiatly gussd that th primtal data ould b fit by th modifid formula b d = a 5 ( b ) H st his guss to th primtalist who ofirmd that it did fit th data with a propr hoi of th ostats a ad b. Pla ow dd to fid a way to driv th formula iludig th valus for th ostats. Pla was a formr studt of Kirhhoff s ad had dvlopd a rsarh program o applyig th Sod aw of Thrmodyamis to problms at a tim wh th law was ot widly applid. Pla's appliatio of thrmodyamis ovid him that th avrag rgy alulatio was iorrt. Pla didd that th rgy of th harmoi osillator stats should b rprstd by ε whr is a itgr,,,...

12 Pla's argumts for this statmt ar byod th sop of this ours. Th itrstd studt a fid a disussio i Th Quatum Physiists ad a Itrodutio to Thir Physis by William H. Coopr. W ow alulat th avrag rgy for a rgy stat usig this rlatioship btw rgy ad frquy. h ε Th bottom summatio a b foud i a math hadboo ad is giv by h h W a us th followig Calulus tri to hlp valuat th top summatio d d Whr w will dfi to b h h h d d d d h d d

13 - h h h - h W ow substitut our rsults ba ito avrag rgy quatio ad obtai ε h h h h Pla ow multiplid th avrag rgy by th lassial dsity of stats to obtai his radiatio formula of 8π g ε d du 3

14 Pla s Radiatio aw (Corrt Rsult) d π 3 or quivaltly d 5 πh W s that th potial i th domiator prvts th ultra-violt atastroph by ausig th itsity to dras potially at high frquis. W also a show that Pla's rsult giv th Rayligh - Ja's rsult for low frquy by padig th potial futio as follows for h d 3 π π Pla was awar that a lassial harmoi osillator should hav a otiuous rgy distributio. Howvr, h thought that th ultra-violt atastroph might b du to a ovrg problm with th itgratio. Thus, h mad th rgy lvls disrt so that th avrag rgy alulatio ivolvd a sum. H itdd to obtai th otiuous rgy distributio of th lassial harmoi osillator by lttig th paramtr h approah zro i his fial rsult. Howvr, lttig h go to zro rsultd i th ultra-violt atastroph. Thus, h had to b lft

15 fiit. A llt fit of primtal blabody sptra ould b obtaid usig th valu for h of h 4 Vm This ostat is alld Pla's ostat!! FACT: Pla was a lassial physiist ad vr blivd that this ostat had ay grat physial sigifia. I partiular, h did't bliv that th osillator's rgy lvls wr atually quatizd! W ow ow that th pross rquirs quatum mhais to dsrib. Pla had usd th wrog rgy statistis with th wrog rgy to gt th right aswr! A youg patt lr amd Albrt Eisti had a muh diffrt viw of th ostat ad would ma it tral to his solutio of a problm that would wi him th Nobl Priz!