Planks Black body spectrum equation is given by,
Planks Spectral energy density, µ(ʋ,T)
The factor µ(ʋ,T) is called the Black body spectrum
♦DERIVATION
Consider a box of side ‘l’
The resonating modes of the electronegative radiation inside the box have the frequency, [docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p2-2.odt” comments=”true”]
The energy of the proton is given by planks hypothesis as[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p3.odt” comments=”true”]
For a three dimensional box the energy expression will be[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p4-1.odt” comments=”true”]
The total energy, U, in a 1-D box is[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p5-2.odt” comments=”true”]
Therefor in a 3-D box the energy U will be[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p6-1.odt” comments=”true”]
Assuming the ni s to be continuous variables the summations can be replaced by integrals (fermi approximation)
Thus the total energy U is given by[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/plnk-last.odt” comments=”true”]
U(ʋ,T) is called the Black body spectrum and the equation is called Planks Black body spectrum equation
where
This spectral density function has units of energy per unit wavelength per unit volume.
Total energy
Evaluation of the integral to obtains the total energy inside the box yields