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