; TeX output 1998.01.17:1522nWn{aH{DtqGcmr17ModernBPhuNysicsƦXQ ff cmr12Final/ExamLqK`y cmr10Name:qDŽff\jStudentUUI.D.:qDŽff\`1*nWn^Short/Answer(10ptseach)Answer?allproblemsinasthoroughdetailaspGossible.Besuretoincludeallyourwork.Partialcredit willUUbGegiveneveniftheanswerisnotfullycorrect. 81.DerivefStefan'slawfromPlanck'slaw."6Usingthatresult,_obtainanexpressionfortheStefan-BoltzmannconstantUU b> cmmi10.intermsofknownphysicalconstantsanddetermineitsvqalue.g 82.Thev7Ei elT*owerv7inParisis300mtall.'hWhatisthefractionalgravitationalredshiftduetothiselevqationdi erence?\E 83.DeterminethechargeintherestmassofasystemconsistingofaprotonandanelectronasthetwoparticlesUUcombinetoformahydrogenatom.qTheionizationenergyofhydrogenis13.6eV. 84.Calculateo.thedensityofanob8jectofonesolarmasswhoseradiusisthecriticalSchwarzschildradius.CompareUUthisdensitywiththenucleardensityofapproximately2:38 !", cmsy1010^ٓRcmr717xkPg[=m^3|s.`2CnWnl)]A/PartialListofF4undamentalConstantsPRˍiMConstantySymbGol'ApproximateUUV*aluecMff'fdSpGeedUUoflightinvqacuumBcg3:00810^8|sm/s PermeabilityUUofvqacuum*0g12:6810^ O!cmsy77 tH/mPermittivityUUofvqacuum&0g8:85810^12F/m fMagneticUU uxquantumqֵ0C=  0ercmmi7hK&feǾ2eg2:07810^15WbElectronUUmassmeg9:11810^31kgProtonUUmassmpg1:673810^27kgNeutronUUmassOٵmng1:675810^27kgProton-electronUUmassratio텍ƼmO \cmmi5pƼŭfe Ko( meg1836`3TnWnqd{ModernBPhuNysicsƦFinal/ExamV7qName:qDŽff\jStudentUUI.D.:qDŽff\`1 nWn?oiODtHGcmr17AnswYerKeyforExamffV|GGffHGcmss17AGffffffV|(Short/Answer(10ptseach)Answer?allproblemsinasthoroughdetailaspGossible.Besuretoincludeallyourwork.Partialcredit willUUbGegiveneveniftheanswerisnotfullycorrect. 81.DerivefStefan'slawfromPlanck'slaw."6Usingthatresult,_obtainanexpressionfortheStefan-BoltzmannconstantUU.intermsofknownphysicalconstantsanddetermineitsvqalue."V cmbx10Answ9er:FΈTheUUradiance 82.Thev7Ei elT*owerv7inParisis300mtall.'hWhatisthefractionalgravitationalredshiftduetothiselevqationdi erence?rAnsw9er:絍<$`wfe f8 (֍iU"-=<$Kg[HKwfe@ (֍Jcr25w=<$K(9:8m=s^2|s)(300m)KwfeK@ (֍PR(3810r8|sm=s)r2Sֲ=3:2781014MЍ 83.DeterminethechargeintherestmassofasystemconsistingofaprotonandanelectronasthetwoparticlesUUcombinetoformahydrogenatom.qTheionizationenergyofhydrogenis13.6eV.Answ9er:FΈThelrestmassoftheprotonis1:67264910^27kg;thatloftheelectronis9:10953410^31kg./WhenЁtheelectronandprotoncombinetoformahydrogenatom,Ltheionizationenergyof13.6/eVisreleasedasultravioletradiation.qTherestmassofthehydrogenatomisthereforesmaller/thanUUthesumoftheelectronandprotonrestmassesbytheamount|M3=<$K(13:6eV )(1:6810^19Jh=eV)Kwfeua (֍b(3810r8|sm=s)r2}=2:4281035kPgMЍ 84.Calculateo.thedensityofanob8jectofonesolarmasswhoseradiusisthecriticalSchwarzschildradius.CompareUUthisdensitywiththenucleardensityofapproximately2:3810^17xkPg[=m^3|s.Answ9er:FΈTheumassofthesunis2N10^30xkPg[ٲ.PTheuuniversalgravitationalconstantis6:67N10^11Ne/m^2|s=kPg[ٟ^2L./TheUUSchwarzschildradiusofonesolarmassis絍.Rcٓ=<$2(2810^30xl2`g[ٲ)(6:6710^11NOm62=kPg^2L)wfeI (֍8 V(3810r8|sm=s)r2⍍ٓ=2:968103|sm3kPm/Ifthesuncollapsedtoasphereof3kmradiuswithoutlossofmass,itwouldthenbGeablackhole./TheUUmassdensitywouldbGe|=<$KMKwfe ʪ (֍^9V@=<$2810^30xkPgKwfeT! ( 33433&fev3 PH)(2:96810r3|sm)r3\=1:8481019xkPg[=m3`2 nWnl)]A/PartialListofF4undamentalConstantsPRˍiMConstantySymbGol'ApproximateUUV*aluecMff'fdSpGeedUUoflightinvqacuumBcg3:00810^8|sm/s PermeabilityUUofvqacuum*0g12:6810^7 tH/mPermittivityUUofvqacuum&0g8:85810^12F/m fMagneticUU uxquantumqֵ0C= hK&feǾ2eg2:07810^15WbElectronUUmassmeg9:11810^31kgProtonUUmassmpg1:673810^27kgNeutronUUmassOٵmng1:675810^27kgProton-electronUUmassratio텍ƼmpƼŭfe Ko( meg1836`3(nWnqd{ModernBPhuNysicsƦFinal/ExamV7qName:qDŽff\jStudentUUI.D.:qDŽff\`1MnWn^Short/Answer(10ptseach)Answer?allproblemsinasthoroughdetailaspGossible.Besuretoincludeallyourwork.Partialcredit willUUbGegiveneveniftheanswerisnotfullycorrect. 81.DeterminethechargeintherestmassofasystemconsistingofaprotonandanelectronasthetwoparticlesUUcombinetoformahydrogenatom.qTheionizationenergyofhydrogenis13.6eV.g 82.Thev7Ei elT*owerv7inParisis300mtall.'hWhatisthefractionalgravitationalredshiftduetothiselevqationdi erence?\E 83.Calculateo.thedensityofanob8jectofonesolarmasswhoseradiusisthecriticalSchwarzschildradius.CompareUUthisdensitywiththenucleardensityofapproximately2:3810^17xkPg[=m^3|s. 84.DerivefStefan'slawfromPlanck'slaw."6Usingthatresult,_obtainanexpressionfortheStefan-BoltzmannconstantUU.intermsofknownphysicalconstantsanddetermineitsvqalue.`2'nWnӍl)]A/PartialListofF4undamentalConstantsPRˍiMConstantySymbGol'ApproximateUUV*aluecMff'fdSpGeedUUoflightinvqacuumBcg3:00810^8|sm/s PermeabilityUUofvqacuum*0g12:6810^7 tH/mPermittivityUUofvqacuum&0g8:85810^12F/m fMagneticUU uxquantumqֵ0C= hK&feǾ2eg2:07810^15WbElectronUUmassmeg9:11810^31kgProtonUUmassmpg1:673810^27kgNeutronUUmassOٵmng1:675810^27kgProton-electronUUmassratio텍ƼmpƼŭfe Ko( meg1836`3"nWnqd{ModernBPhuNysicsƦFinal/ExamV7qName:qDŽff\jStudentUUI.D.:qDŽff\`1'nWn^Short/Answer(10ptseach)Answer?allproblemsinasthoroughdetailaspGossible.Besuretoincludeallyourwork.Partialcredit willUUbGegiveneveniftheanswerisnotfullycorrect. 81.DerivefStefan'slawfromPlanck'slaw."6Usingthatresult,_obtainanexpressionfortheStefan-BoltzmannconstantUU.intermsofknownphysicalconstantsanddetermineitsvqalue.g 82.DeterminethechargeintherestmassofasystemconsistingofaprotonandanelectronasthetwoparticlesUUcombinetoformahydrogenatom.qTheionizationenergyofhydrogenis13.6eV. 83.Thev7Ei elT*owerv7inParisis300mtall.'hWhatisthefractionalgravitationalredshiftduetothiselevqationdi erence?\E 84.Calculateo.thedensityofanob8jectofonesolarmasswhoseradiusisthecriticalSchwarzschildradius.CompareUUthisdensitywiththenucleardensityofapproximately2:3810^17xkPg[=m^3|s.`2'nWnl)]A/PartialListofF4undamentalConstantsPRˍiMConstantySymbGol'ApproximateUUV*aluecMff'fdSpGeedUUoflightinvqacuumBcg3:00810^8|sm/s PermeabilityUUofvqacuum*0g12:6810^7 tH/mPermittivityUUofvqacuum&0g8:85810^12F/m fMagneticUU uxquantumqֵ0C= hK&feǾ2eg2:07810^15WbElectronUUmassmeg9:11810^31kgProtonUUmassmpg1:673810^27kgNeutronUUmassOٵmng1:675810^27kgProton-electronUUmassratio텍ƼmpƼŭfe Ko( meg1836`3,ȠnWnqd{ModernBPhuNysicsƦFinal/ExamV7qName:qDŽff\jStudentUUI.D.:qDŽff\`10nWn^Short/Answer(10ptseach)Answer?allproblemsinasthoroughdetailaspGossible.Besuretoincludeallyourwork.Partialcredit willUUbGegiveneveniftheanswerisnotfullycorrect. 81.Calculateo.thedensityofanob8jectofonesolarmasswhoseradiusisthecriticalSchwarzschildradius.CompareUUthisdensitywiththenucleardensityofapproximately2:3810^17xkPg[=m^3|s.g 82.Thev7Ei elT*owerv7inParisis300mtall.'hWhatisthefractionalgravitationalredshiftduetothiselevqationdi erence?\E 83.DerivefStefan'slawfromPlanck'slaw."6Usingthatresult,_obtainanexpressionfortheStefan-BoltzmannconstantUU.intermsofknownphysicalconstantsanddetermineitsvqalue. 84.DeterminethechargeintherestmassofasystemconsistingofaprotonandanelectronasthetwoparticlesUUcombinetoformahydrogenatom.qTheionizationenergyofhydrogenis13.6eV.`21ǠnWnӍl)]A/PartialListofF4undamentalConstantsPRˍiMConstantySymbGol'ApproximateUUV*aluecMff'fdSpGeedUUoflightinvqacuumBcg3:00810^8|sm/s PermeabilityUUofvqacuum*0g12:6810^7 tH/mPermittivityUUofvqacuum&0g8:85810^12F/m fMagneticUU uxquantumqֵ0C= hK&feǾ2eg2:07810^15WbElectronUUmassmeg9:11810^31kgProtonUUmassmpg1:673810^27kgNeutronUUmassOٵmng1:675810^27kgProton-electronUUmassratio텍ƼmpƼŭfe Ko( meg1836`36;nG "V cmbx10HGcmss17DtHGcmr17XQ ff cmr12DtqGcmr17 !", cmsy10 O!cmsy7 b> cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7: