A tutorial for laboratory determination of Planck s constant from the Planck radiation law

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1 A tutorial for laboratory dtrmination of Planck s constant from th Planck radiation law Adam Usman, John Dogari, M. idwan Enuwa and sa Sambo Dartmnt of Physics, Fdral Univrsity of Tchnology, P. M. B. 076, Yola, Adamawa Stat, 6000, Nigria. [email protected] (civd May 009; acctd May 009) Abstract Dtaild discussions ar rsntd on th dtrmination of Planck s constant from th Planck radiation law. Th laboratory stu consists of a low cost lam and a hotodiod such that a light filtr of known wavlngth is introsd btwn th two ics of aaratus. Th lam filamnt is th aroximat blackbody of which radiation is govrnd by th Planck s law of radiation. For th uross of didactics, it is xlaind that laboratory rcautions will nabl accurat data. Howvr, if th govrning quation is subjctd to surious aroximations, rrors will b invitabl in th comutation. This was th situation bfor. Now, whn th stu is in oration, a hnomnon of rsistanc-tmratur filtr ffct is obsrvd. Data collctd hav bn subjctd to intrsting analyss, which can b usd to introduc studnts into numrical xrimnts. Usually, a air of intnsity valus calculatd from th obsrvd hnomnon, would b rquird. atio of th intnsitis is thn usd in th xact Planck s radiation quations lading to a transcndntal xonntial quation. Solutions of this quation subsum th rcis valu of th Plank s constant. Kywords: Planck s constant dtrmaintion, numrical xrimnt, hotomratur. sumn A continuación s rsnta un gruo d discucions dtalladas sobr la dtrminación d la constant d Planck qu aarc n la ly d radiación dl mismo nombr. En un laboratorio s dison d una instalación d bajo costo qu consta d una lámara y un fotodiodo, d tal forma qu un filtro d luz d longitud d onda conocida s intrusto ntr las dos izas dl aarato. El filamnto d la lámara s coomorta como un curo ngro cuya radiación stá gobrnada or la ly d radiación d Planck. Para fins didácticos, s xlica qu s obtinn datos aroximados como una mdida d rcaución n l laboratorio. Sin mbargo, si la cuación gobrnant stá sujta a aroximacions surias, los rrors s roagarán invitablmnt n los cálculos. Esta fué una situación antrior. Ahora, cuando s on n marcha sta nuva configuración, un fnómno d filtración ntr la rsistncia y la tmratura s obsrvado. Los datos rcolctados han sido sujtos a análisis intrsants, los cuals udn sr utilizados ara introducir a los alumnos n l studio d los xrimntos numéricos. Usualmnt, srán rquridos un ar d valors d intnsidad calculados a artir dl fnómno obsrvado. La razón d las intnsidads s ntoncs usada n las cuacions d radiación d Planck llvándonos a una cuación trascndntal xonncial. Las solucions d sta cuación nos rovn d un valor rciso ara la constant d Planck. Palabras clav: Dtrmainción d la constant d Planck, xrimnto numérico, fototmratura. PACS: 0.50.Pa, 06.0Jr SSN NTODUCTON n all quantum hnomna, it aars that th Planck s constant, h, is vr rsnt. A notabl xrimntal dtrmination of, h, may b attributd to th work of Millikan during th yars in th riod of 905 to 96. With sohiscatd aaratus, Millikan s final rsult was (5.57 ± 0.0) 0 - Js []. This was basd on th Einstin intrrtation of th hotolctric ffct. On might surmis this as th rason for th xistnc of svral form of hotolctric ffct xrimnts in ordr to dtrmin, h. n th rsnt days fforts, thr ar mor sohiscatd sts of aaratus [], that ar basd on othr rincils. t is, howvr, intrsting to notic that th dtrmination of, h, in all xisting xrimnts is dndnt on othr fundamntal constants []. This is also th trnd to b obsrvd in many undrgraduat laboratory xrimnts [, ]. n this work, too, th Boltzmann s constant, k B, is th fundamntal constant associating with, h. W not with nthusiasm, that, thr is an absolut intrsting laboratory xrcis for th dtrmination of k B [5]. n all undrgraduat laboratory xrciss on, h, th mhass ar on rcautions in th functions of th aaratus. Whil this is good training for th studnts, it is insufficint. Emhass ar also ncssary on th choics of Lat. Am. J. Phys. Educ. Vol., No., May htt://

2 Adam Usman, John Dogari, M. idwan Enuwa and sa Sambo mthods of analysis of data scially whn aroximations ar to b usd. On of th aims of this rsntation is an xosition of wrong rsults owing to mislading aroximations of th govrning Planck s radiation quation, []. Anothr aim is to furthr tst th hnomnon of rsistanc-tmratur filtr ffct, TFE, [6]. Th TFE is du to th hysical imlications that th filtr usd in th xrimntal stu has th ability to filtr tmratur. This tmratur is crtainly th hototmratu consqunt to th hotocurrnt dtctd by th hotodiod usd. n Sction, th mannrs by which th Planck radiation quation is usd is xlaind in mor dtails than that was givn in [6]. Sction will giv th laboratory stu of th xrimnt with furthr xlanation of th TFE. All dtails of th numrical xrimnts and analyss ar rsntd in Sction V. Discussions of rsults ar in Sction V. Sction V is for conclussions.. THEOY Th Planck radiation law govrns th nrgy distribution in blackbody radiation [7]. n trms of frquncy of th radiation, th law can b statd as ρ 8π, () c ( ν, T ) dν { x( kt ) } dν whr ρ, in units of J/m /s -, is th radiation nrgy r unit volum, and r unit frquncy intrval, dν, at th tmratur, T, c is th sd of light, k B is th Boltzmann s constant and h is th Planck s constant. For quation () to b alicabl it has to b xrssd in th standard units of intnsity, which ar W/m. Multilying both sids of () by c/ν givs c ( ν, T) ρ( ν, T) ν 8πν h { x( kt) }, c whr will b usd to dnot intnsity and i will dnot lctric currnt in what follow. Units of th quation ar now W/m as givn by quation (). For a fixd frquncy, ν, at diffrnt tmraturs T j and T l th ratio of any two intnsitis j (T j ) and l (T l ) is ( ) ( ) ( ν ) j Tj x h ktl, T x h kt ( ν ) l l j whr j l,,, t should b obsrvd that quation () is xrssibl as () () l l x( ktj) x( ktl) + 0, (5) j j so that for masurd valus of intnsitis j and l, and known valus of T j, T l and ν, quation (5) is a form of transcndntal quation which dos not hav a closd form of solution for h/k to b found [9]. t is, howvr, an intrsting xrcis to aly anyon numrical rocdur such as th Nwton s Mthod or th fixd-oint itration mthod [8]. Th Nwton s Mthod was attmtd in this tutorial. n th visibl rang of frquncis usd, th aroximat form of quation () basd on >> kt j so that x(/kt j ) >> is [] j ln. l k Tl T j sults from alications of quations (5) and (6) will b discussd in mor dtail in th following sctions. Th laboratory stu involvs masurmnts of sris of voltags V j with thir corrsonding lctric currnt, i j, of a light bulb filamnt as shown in Fig.. To dtrmining th intnsitis j and tmratur T j on rquirs to us th Stfan-Boltzmann law which can b xrssd as [] (6) W P/A σt, (7) whr σ is th Stfan-Boltzmann constant and A is th surfac ara of th filamnt nclosd in th bulb, and P is th owr mittd. FLTE Pd sistor FGUE. Laboratory stu of lctric circuit. A and E should b stabilizd sourcs and not ordinary battris which drain vry scond. S txt for furthr xlanation. γ f a owr law of th form T is assumd, and an mirical rlation btwn rsistanc of th filamnt,, and tmratur T, through th owr dissiatd can b xrssd as P i AσT β, (8) in which β and γ ar constants, and from quation (8) th mirical rlation of and T is [] γ D E Lat. Am. J. Phys. Educ. Vol., No., May htt://

3 T A tutorial for laboratory dtrmination of Planck s constant from th Planck radiation law γ ( γ ) T, (9). () 0 0 whr 0 dnots th filamnt rsistanc at tmratur T 0. Th room tmratur has bn usd as T 0 for which 0 was masurd using a MCONTA digital rsistanc mtr, [6]. Using quation (8) in (9) yilds whr ( i ) ln C + γ ln ln, (0) 0 γ 0 T C Aσ, (0a) so that a lot of ln( i ) on th vrtical axis against ln( ) on th horizontal axis is xctd to giv a straight lin in ordr to dtrmin th owr, γ, using th last squars fit. On should notic that th rvious works rortd hav bn basd on th aroximation so far xlaind [, 9, 0]. Also, th circuit givn in Fig. is similar in oration to that was usd in [], imlying that similar raw data would b rcordd. Howvr, in our rsnt tutorial, w hav alid TFE which is du to filtr ffct on th mittd owr. t is ncssary to xlain th TFE at this oint. Figur consists of two circuits: th hat mitting circuit and th dtctor circuit. Although th air of circuits orats in comlt analogy to thrmionic convrsion circuits of hat to lctricity [], th latr has th hotodtctor fundamntally of which function is to dtct th amount of currnt that is roortional to th filtrd hat. n th filtring rocss, thrfor, thr is som contnt of tmratur which is roortional to th dtctd currnt. Th hotoowr contnt i, whr i and dnot hotocurrnt and hotorsistanc, should hav th rlation of Stfan-Boltzmann law as xrssd by quation (8) for th sam owr valu, γ, alrady dtrmind. That is, for th dtctd currnt, i, th mirical rlation with and T is P i AσT β, () t should b noticd that i ar rcordd simultanously with V j and i j in th raw data. Equation () which ariss from TFE nabls us to calculat and T. For xaml, by taking th ratio of quation (8) and () on gts P P i i γ γ. () By xrssing T in th sam form similar to quation (9), T ( / 0 )T 0 and thn taking th ratio of this xrssion to quation (9), T is obtaind in trms of i, i and T as T γ ( γ ) T. () t is rtinnt to ralis that th hotointnsity of th filamnt du to TFE for anyon masurd st of V j, i j and i is i /A. Anothr st of raw data valus V l, i l and i l givs hotointnsity l il l /A whr and l ar calculatd using quation (). With th fixd valu of frquncy (scifid for th filtr), ν, xrssions of th alicabl Planck radiation law givn by quation () can b writtn for and l. atio of th hotointnsitis takn as in quation () would lad to l l x( kt ) x( ktl ) + 0, (5) whr T and T l ar obtainabl from quation (). Thortically, on dducs that >> kt is an imrovd aroximation ovr >> kt j, if T j > T, so that x(/kt ) >> is, by imlication, mor rliabl. Now, a form of quation (6) whn TFE is invokd is ln l k Tl T. (6) t should b obsrvd that in ithr quation (5) or quation (6), th filamnt surfac ara A is not rquird []. f it is ncssary, A can b dtrmind from C as givn by quation (0a). Equation (6) is an imrovd vrsion of quation (6) as givn in rf. []. Howvr, th two quations do not giv rasonabl rsults as would b sn shortly. Advancd dgr studnts would find it stimulating to attmting solutions of quations (5) and (5) using anyon of svral numrics [8]. Evn thn, thr ar roblms. On roblm concrns th larg valu of x(/kt j ) or x(/kt ). Anothr on is that du to th rsnc of xonnts; ithr quation has svral solutions dnding on starting inut valus whn th Nwton s Mthod is alid. n this wis a siml Mathmatica statmnt of Findoot hls a lot. Ovrall, analyss show that quation (5) contains an xact valu of h, th Planck s constant. Equation () lads to Lat. Am. J. Phys. Educ. Vol., No., May htt://

4 Adam Usman, John Dogari, M. idwan Enuwa and sa Sambo. LABOATOY SETUP OF THE EXPE- MENT As indicatd in Fig.. most of th ics of aaratus ar common in many undrgraduat laboratoris. A is a stabilisd variabl d.c owr sourc, of rang 0.0 V to 50.0 V; B and C ar MCONTA digital multimtrs for masurmnts of voltag and currnt. D hans to b an analog microammtr as it is an imortant rcaution to avoid saturation currnts which may likly occur if masurmnt of dtctd currnts ar in miliamrs. E is a PHYWE stabilisd a.c./d.c owr sourc of which th d.c sourc was oratd at a constant valu of.0 V to biasing th S Photodtctor which was chosn bcaus of its larg ara and flxibility to various otical filtr frquncis u to a ak wavlngth of 900 nm. Th stu consists of two circuits (s Fig. ): th hat mitting circuit through a 60W light bulb, and th hat (currnt) dtctor circuit ssntially by th S hotodtctor Pd. Th two circuits communicat by on way from th lam to th hotodtctor. Howvr, th dtctd currnt, i, is masurd with a microammtr D and th function of th rsistor,.0 kω, is to dissiat th dtctd owr. Crtainly th circuit (Fig. ) is siml to orat. By incrasing th d.c. voltag, A, in th hat mitting circuit, th filamnt rsistanc,, and consquntly its tmratur T, incras. Th voltag across th filamnt is masurd by B and dnotd as V whil th lctric currnt through it is i masurd by C. Sontanously, Pd, dtcts th hat convrtd to hotocurrnt, i, masurd by D of which valu dnds on th introsing filtr (s Figs. and ). t is thrfor, conformabl to rason that th dtctd currnt has som contnt of rsistanc, and consquntly tmratur T filtrd from and T. This rocss w rfr to as rsistanc-tmratur filtr ffct, TFE, rsulting into quations () and (). V, i, and i ar th hysical quantitis of th raw data shown in tabl ; othr quantitis rcordd ar calculatd as indicatd FGUE. Schmatic drawing of ssntial ics of aaratus:. Conncting cords to variabl Gnral \uros (Phili Harris) owr suly,. Bulb sockt,. Lam filiamnt,. Lam nulb, 5. PHYWE light filtr, 6. Polyvilnyl tub, 7. Conncting cords to biasing voltag (i.., PHYWE stabilzd owr surc), 8. S Photodtctor. 5 TABLE. aw xrimntal data and calculatd filamnt rsistanc, owr, tmratur and hotocurrnt; othr quantitis ar calculatd data du to TFE: hotorsistanc, hotoowr, and hototmratur. Paramtrs usd ar: T 0 (oom tmratur) 0 K, 0 (rsistanc at T 0 ) 6.5 Ω, bulb wattag 60W, tub lngth, L t.0 cm, biasing voltag, V b.0 V, filtr wavlngth, λ f nm. V (V) i (A) V /i (Ω) P i (W) T (K) i (μa) (Ω) P i (W) T (K) t is found that avoiding darkning th laboratory is both rcautionary [] and alicabl to most othr xrimnts in our didactic rogramm. A blacknd-insid olyvinyl tub of diamtr 8.0 cm was usd insid of which th communication of lam bulb and hotodtctor taks lac thus avoiding othr sourcs of intrfring radiation. Amount of currnt dtctd was found to dnd on tub lngth: shortr tub givs mor intnsity of hotocurrnt. Too, th tub fittd xactly into th PHYWE filtrs usd. A siml thrmomtr was usd to masur T 0, th room tmratur, and 0, th rsistanc at T 0, was masurd using MCONTA Digital Multimtr. T 0 and 0 ar rquird in quation (9). Th data rcordd in tabl ar for th PHYWE yllow light filtr of wav lngth, λ f nm. Lat. Am. J. Phys. Educ. Vol., No., May htt://

5 A tutorial for laboratory dtrmination of Planck s constant from th Planck radiation law sults ar also rortd hr of data collctd (s tabl ) for PHYWE grn and blu light filrs at th sam T 0, 0 and othr aramtrs rcordd in tabl. Actually, th data discussd hr constitut just on st of a vry larg numbr of sts of data collctd and analysd ovr a riod of two yars u to now. Th ffort has so far bn to diligntly obsrv any drastic dviation from th TFE. V NUMECAL EXPEMENTS n this sction w dtail our numrical rocdurs. W bgin with th dtrmination of γ-aramtr, [] which is clarly by using quation (0). Th rsulting straight lin (s Fig ) from th last squars fit yilds th valu of γ that is subsquntly usd to calculat th variabls givn in tabl. Scifically, th variabls ar T from quation (8), from quation (), T from quation () and P i. Th significancs of quation (0) includ dtrmination of surfac ara of th hat mitting filamnt through th aramtr C. ln(p ) ln (P ).9989 ln ( ) γ ln( ) FGUE. Grah of ln P (mittd owr) against ln (filamnt rsistanc) to dtrmin γ. S tabl for calculations of P and. W first sought to comut th Planck s constant, h, using th rocdur of rfrnc []. By quation (5), th imlication of instruction givn in, [], is that j i so that w hav ln kb Tj T, (7) whr ν c/λ with c maning th sd of light, λ, th wavlngth of filtr in us as givn with th data of tabl, and k B is th Boltzmann constant of which valu usd hr, [], (OP, 999/000), is JK -. n quation (7), it should b notd that T was takn as th rfrnc tmratur of th filamnt so that a lot of (/T j /T ) along th horizontal axis against ln(i /i ) along th vrtical axis should yild a straight lin; th slo of this lin obtaind by th last squars fit should b quivalnt to /k B. Using th data givn in tabl, th lottd oints giv straight lin. n fact th last squars quation is i ln , i Tj T (8) whr j,,,, 9. Th grah is not shown hr for on obvious rason that j i. Th corrct rlation is i /A whr A is th filamnt surfac ara. Th gradint valu 585. from quation (8) yilds h Js. Othr qually carfully collctd data yildd valus in th rang of Js to Js. W nxt usd quation (5) with th incorrct i rlation. Equation (5) would thn b i x BT i x i + BTj i 0. (9) t aars that quation (9) should b solubl with anyon standard numrical mthods [8], in which, h, bcoms th unknown variabl to b found. Howvr, thr ar roblms. First, th rsulting valu in th xonnt, obtaind by valuating ν/k B T j is too larg for any hand calculator. Us of FOTAN 95 givs floating oint; th rogarmms could not b xcutd. t was suosd that th roblm could b ovrcom by writing quation (9) as i i x( x T ) x( x Tj ) + 0, (0) i i so that x /k B. Still du to th xonnt, thr is anothr roblm whn th Nwton s mthod [8], was usd in FOTAN coding. Equation (0) has svral solutions that closly dnd on th inut valus. Hr, w rort most intrsting rsults whn th siml statmnt of Mathmatica Findoot [] was usd (s Andix A). On notics that th Findoot statmnt works by th rincil of Nwton s mthod of solving quations. Th non-uniqu solution still rsists for ithr quation (9) or (0). Howvr, Mathmatica is abl to rturn a solution. By inutting a valu such as which is clos to th standard valu [], on could gt a solution. Onc mor, it was found that th incorrct j i rlation affcts th rsults of quation (9). W now focus attntion on quations (5) and (6) which rsultd from th us of TFE on th hat mittd Lat. Am. J. Phys. Educ. Vol., No., May htt://

6 Adam Usman, John Dogari, M. idwan Enuwa and sa Sambo by th filamnt. With th data in tabl, quation (6) bcoms ln k T T. () Fig. shows th lot of (/T /T ) along th horizontal axis against ln(p /P ) along th vrtical axis. Th last squars lin yildd is P ln P.6609 T () T Th valu obtaind for h hr is Js. 5 x( x) x( x) () Mathmatica is abl to rturn Js with a starting valu of Js. Mathmatica rturns th sam, h, valus for j,,,, 9. Whn th rcirocals of th ratios of intnsitis ar usd, i.., instad of /, on usd /, by corrctly rwriting th rsulting quations th sam rsults wr obtaind. All rocdurs abov wr for th cas of yllow filtr. Data wr collctd and similarly analyzd for grn and blu filtrs. S tabl. TABLE. sults obtaind for Planck s constant using th stu of Figs. and for th thr filtrs for th sam ambint conditions and with th sam numrical xrimnts xlaind in th txt. Filtr Colour Filtr Wavlngth (nm) Planck s constant, h (Js) 0 Blu ± Grn ± Yllow ± ln(p /P ) (/T - /T ) FGUE. Plotting of quation (7) with a linar fit to calculatd data using th TFE. Not that i /A P /A and l i l l /A P l /A, whr A is th filamnt surfac ara. S txt and tabl for furthr xlanation. Equation () is th last squaurs lin. Whn quation (5) was usd th rsults ar satisfactory using th Mathmatica statmnt givn in Andix A. Th analysis is in th following. With th data in tabl, quation (5) bcoms x r BT x + r BT whr r i i in quation (). As an xaml, if j, quation () is 0, () V. ESULTS AND DSCUSSON Basd on th siml Mathmatica Findoot statmnt (s Andix A), our rsults ar givn in tabl. Th rsult obtaind for th blu filtr is of intrst for on rason: th rang of th multimtrs usd did not rmit mor than two data oints, and th microammtr usd for masurmnt of hotocurrnt is analogu. f w usd a digital microammtr, or vn bttr a digital nanoamtr, it would b ossibl to divid th intrval into, at last, fiv or mor data oints. n dd, undr th sam conditions of ambint tmratur, T 0 0K, th Planck s constant valu, h, obtaind for th yllow and blu filtrs is th sam. t could b obsrvd that th quotd absolut rror is tolrabl. (tabl ). V. CONCLUSONS Figurs and dict th laboratory stu usd for th dtrmination of th Planck s constant, h. Hr, th alid law is th Planck radiation quation. Th fforts so far hav bn dirctd towards rsntation of dtaild rocdurs of numrical xrimnts. f laboratory rcautions ar mhasizd for didactics, it is now clar that xrimntal situations abound whr numrical rcautions ar invitabl. W circumvntd systmatic rrors that affctd rvious rsults [, 9, 0]. t was ralizd that rrors rsultd from mislading aroximations in th govrning quation. t was found, for an xaml, that th aroximations aar to b imrovd whn variabls T wr usd in x(/k B T ) >> instad of variabls T j in x(/k B T j ) >>, whr j,,,.., 9; 9 bing th Lat. Am. J. Phys. Educ. Vol., No., May htt://

7 A tutorial for laboratory dtrmination of Planck s constant from th Planck radiation law numbr of data oints in th laboratory masurmnts for th cas of yllow filtr. That is bcaus th mittd tmraturs, T j ar not of qual magnitud with th hototmraturs T. n anothr words if i j i, it is rasonabl to xct T j T. Emirically, w find that T j > T. t would b rcalld that T dfin tmraturs calculatd using quation () which is on of th rsults of TFE alrady xlaind in th txt. T j ar th tmraturs of th mittd hat by th filamnt, which would consquntly b filtrd to roduc T. t was obsrvd that th filtrd currnts (i.., th hotocurrnts) i would always b lss than th mittd currnts, i j, (i.., i j > i ). For final analysis, it was found to b corrctiv by dvloing and solving quation (5) numrically with a siml Mathmatica Findoot (s Andix A). Th quation has svral solutions ach dnding on th inut or starting valu. Alongsid th corrct rlation btwn hotointnsitis,, and hotocurrnts, i, which is i /A; A bing th surfac ara of th filamnt nclosd insid th bulb, w found that (s tabl ) th rcommndd valu of th Planck s constant, h, is on of th sris of th solutions of quation (5). n articular, if it is assumd that th Planck s constant rmissibl for alications is in th rang of Js to Js, thn, our rsults for yllow and grn filtrs hav absolut rror of 0.009% comard to th standard valu, [], (OP, 999/000). For th blu filtr, th absolut rror is 0.000%. ACKNOWLEDGEMENTS W thank A. A. Makind, th chif tchnologist of Physics Dartmnt, FUTY, for lnding us his S Photodtctor and his frqunt tchnical assistanc. EFEENCES [] Ksing,. G., Th masurmnt of Planck s constant using th visibl hotolctric ffct, Eur. J. Phys., 9-9 (98). [] azt, A., Houssin, O., and Basti, J., A dtrmination of Planck s constant from radiomtric masurmnts, Mtrologia, (006). [] Yaakov, K., Photolctric ffct xrimnt with comutr control and data acquisition, Am. J. Phys. 7, 9-9 (006). [] Gracila B, and Alfrdo J, Planck s constant dtrmination using a light bulb, Am. J. Phys. 6, 89 (996). [5] Shustff, M., t. al., Masuring Bolzmann s constant with a low-cost atomic forc microsco: An undrgraduat xrimnt, Am. J. Phys. 7, (006). [6] Usman, A., and Dogari, J., A didactic not for accurat laboratorydtrmination of Planck s constant from th Planck radiation thory,, 95-0, (006). [7] Hnry, S. and John,. A., ntroduction to Atomic and Nuclar Physics, 5 th d., (Chaman and Hall, USA, 989). [8] Curtis, F. G., and Partrick, O. W., Alid Numrical Analysis, 5 th d., (Addison-Wstly, 99). [9] Crandall,. E. and Dlord, J. F., Minimal aaratus for dtrmination of Planck s constant, Am. J. Phys. 5, 90 (98). [0] Dryzk, J and ubnbaur, K, Planck s constant dtrmination from black-body radiation, Am. J. Phys. 60, 5 (99). [] Hatsooulos, G. N., and Kay, J., Masurd Thrmal Efficincis of a Diod Configuration of a Thrmo Elctron Engin, J. Al. Phys. 9-5 (958). [] OP 999/000 Diary, commndd Valus of hysical Constants and Convrsion Factor. [] Gorg B., Mathmatica: Th Studnt Book (Addision-Wsly, USA, 99). APPENDX A Essntial statmnt of Mathmatica Findoot ar givn hr as quation (A). Data for comutation ar obtaind from tabl. W us th aramtrs of quation (). Answrs rturnd by Mathmatica ar numbrd. E^ (c*x) ((/)*(E^ (c*x))) + ((/).0) /. {c 7.08 * (0^7), c.66 * (0^7),.8968 * (0^ -),.8 * (0^ -)} Findoot[% 0, {x, * (0^ -),. * (0^ -,), 7. * (0^ -)}], (A) ( 0 7 x) ( x) x.7 + x, (A) {x }. (A) Th final answr is givn by quation (A). t should b noticd in th Mathmatica statmnt [], that Js is th starting valu that was chosn and th othr two valus corrsond to an intrval within which rsults ar dsird. Aftr giving a notic of accuracy Mathmatica rturns only on valu as th answr givn by quation (A). Equation (A) is th transcndntal xonntial quation () rturnd by th Mathmatica statmnt. Lat. Am. J. Phys. Educ. Vol., No., May htt://