QMGreensite_merged

2.1. BLACK-BODYRADIATION 23

Theenergydensityofradiationinthebox,asafunctionoffrequency,iseasily
workedoutusingtheequipartitionprincipleofstatisticalmechanics.Thetotalenergy
is

Erad = no.ofdegreesoffreedom×

1

2

kT

= 2 ×(no.ofstandingwaves)×

1

2

kT (2.1)

wherekisBoltzman’sconstantandT isthe temperatureof thebox. Anelectro-
magneticfieldinaboxcanbethoughtofasasuperpositionofaninfinitenumberof
standingwaves;the”degreesoffreedom”aretheamplitudesofeachdistinctstanding
wave. Thefactorof 2 comesfromthefactthateachstandingwavecanbeinoneof
twopossiblepolarizations.
Aswewillseeinalaterlecture,thenumberofstandingwavesthatcanexistin
acubicalboxofvolumeV,forfrequenciesintheinterval[f,f+∆f],is

N(f)∆f=V

4 π

c^3

f^2 ∆f (2.2)

Thentheenergyofradiationinthisrangeoffrequencieswillbe

∆Erad= 2 N(f)∆f×

1

2

kT=

4 πkTf^2

c^3

V∆f (2.3)

TheenergydensityperunitfrequencyE(f,T)istherefore

E(f,T)≡

∆Erad

V∆f

=

4 πkTf^2

c^3

(2.4)

whichisknownastheRayleigh-Jeanslaw.
TheRayleigh-Jeanslawcanbetestedbymakingaholeinthebox,andmeasur-
ingtheintensityofradiationemittedfromtheboxasafunctionoffrequency;this
intensityisdirectlyproportionaltoE(f,T).Radiationfromasmallholeinacavity
isknownas”black-bodyradiation”,becauseanyradiationfallingintotheholeisnot
reflectedoutthehole, butisultimately absorbedbythewalls. Theexperimental
result,compared tothe prediction, isshown inFig. [2.1]. Theorydisagrees with
experimentathighfrequencies. Infact, itisclearthattherehadtobesomething
wrongwiththeory,becausethetotalenergyispredictedtobe

Erad = 2 ×(no.ofstandingwaves)×

1

2

kT

= ∞ (2.5)

simplybecausetherangeoffrequenciesisinfinite,sothereisaninfinitenumberof
differentstandingwavesthatcanbesetupinthebox. Theenergyofaboxisfinite