18.1 a) calculate the total electromagnetic energy inside an oven of volume 1m heated to a temperature of

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Tobias Townsend
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1 . a) calculate the total electromagnetic energy inside an oven of volume m heated to a temperature of F a se equation. 6 F plug in..9 J a Jm b) Show that the thermal energy of the air in the oven is a factor of approximately larger than the electromagnetic energy hermal energy nr λ which is..969 times l arg er than + J ( a). a) Calculate for the Earth, assuming the Earth to be blacbody. max From equation. λ.9 m m 9 max b) Calculate the temperature at which the human eye is most sensitive to a wavelength of nm.9 m m.. a) Find the frequency at which the radiant energy per unit frequency interval of a blacbody is a maximum. How does this compare with the frequency at which the radiant energy per unit wavelength interval is maximum? From equ.

2 h u() v C e dv( u) the occurs at max dv d e + e e e e e which leads to h.. h c this is different from v calculated through λ max.96 h b) Find the frequency at which the cosmic bacground radiation is a maximum and compare your results with that shown in Figure.. Assume that.... S max

3 .) Assume that the radiation from the sun can be regarded as blacbody radiation. he radiant energy per wavelength interval has minimum at nm. a) Estimate the temperature of the sun. b) Calculate the total radiant power emitted by the sun. (he radius of the sun is approximately m λ max nm 9 Plan' s cons tan t h 6.66 Speed of light c m a) emperature of the sun? We now that λmax emperature of the sun 6 b) otal radiant power of sun e? emperature of the sun 6 Radius of the sun r m Stefan-oltzmann Constant δ.6 wm We now that e δτ for sun e δτ r e.6 e.9 e.6 e.6 otal radiant 6 ( wm ) ( 6) ( m). watts (.9) megawatts power emitted by sun e.6 megawatts

4 .) a) Calculate the number of photons in equilibrium in a cavity of volume m held at a temperature K b) Compare this number with the number of molecules the same volume of an ideal gas contains at SP olume of the cavity m emperature of the Cavity Standard emperature olume at SP.d m.l a) umber of photons? We now that Ν v Ν v Ν. where x K Integrated value of x e x K. dx. K x e. dx. umber of photons. x.

5 b) umber of photons. (perm ) of photons A? of molecules At SP.dm.m m contains 6. contains 6. contains 6.. molecules umber of molecules.6 ( perm ) of photons of molecules ( perm ) ( perm ) of photons. of molecules..6

6 .6) assume that the universe is spherical cavity with radius 6 m and temperature.k. How many thermally excited photons are there in the universe? se equation (.6) ( ).) the chemical potential of a boson gas at temperature is approximately -.. Determine the mean number of bosons f ( ε ) in a single particle states having energies of (a), (b)., (c)., (d)., (e). f ( ε ) ( ε µ ) e ε as an example ε µ (. ). we now f ( ε ).. e e.96 others followthe above approach!

7 .) se the result of problem. to show that a) z K, Solution: Result of problem. is () K K this valuein eq put c h K c h K,, b) ( ) ( ). (),, µ µ µ K this valuein eq put K from K c)

8 () ( ) ( ) () S K S eq in and of values Put a problem From b problem from S d)

9 z P + We nowthat z Put this valuein eq P P P We now P, ( ) ( ) ( ) ( ) ( ) ( ), Put this valuein eq ( ) ( ) () ( ).) a) Find the chemical potential of a ilomole of He gas at SP. Express your answer in joules and in electron-volts solution: umber of Helium ( He ) atoms at SP ilomole olume of ilomole of He Standard emperature at SP. dm 6. 6 Mass of He atom m

10 Chemical Potential µ? We now that m µ h 6.6. µ ( 6.66 ) µ.6 J. µ.9ev b) se Equation (.) to show that the mean occupancy of a single particle states having energy solution: is. at SP Energy ε Chemical Potential µ. Mean Occupancy of a single particle f ( ε )? From eq (.) f ( ε ) e( ε µ ) +. e.9 e 69.6 f. ( ε ) Mean occupancy of a sin gle particle f ( ε ).

11 . a) an ideal boson gas consists of He atoms whose ose temperature is. K. Find the boson concentration, the number of bosons per cubic meter. se equation. m.6 h m.6 h.6. inm (. ) m. ( 6.6 ). b) What percentage of the bosons are in the ground state at temperature of - K? use equation (.) In ose-einstein condensation experiment, rubidium - atoms were cooled down to a temperature of nk. he atoms were confined to a volume of approximately m umber of rubidium- atoms emperature nk 9 olume m Mass of rubidium- atom a) Calculate the ose temperature?

12 We now that h m.6.. ( 6.66 ).6 6 osonemperature b) Determine how many atoms were in the ground state at nk emperature otal of atoms 6 9 oson emperature.6 6 We now of atoms in ground state? o o 9.9 [. ] of atoms in ground state (c) Calculate the ratio, where nk and where the ground state energy ε is given by ε ε h m emperature nk

13 Energy ε h Mass of rubidium atom. olume m K Ratio? ε h ε. 6. ε.. ( 6.66 ) ( ).6. K. K.6 K ε K Ratio ε 9

PHYSICS 250 May 4, Final Exam - Solutions

PHYSICS 250 May 4, Final Exam - Solutions Name: PHYSICS 250 May 4, 999 Final Exam - Solutions Instructions: Work all problems. You may use a calculator and two pages of notes you may have prepared. There are problems of varying length and difficulty.