## Derivation Of Plancks Radiation Law

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# Derivation of Planck’s Radiation law

Planck considered the black body radiations (in the hohlraum) to consist of linear oscillators of molecular dimensions and that the energy of a linear oscillator can assume only the discrete values

0,hv,2hv,3hv....nhv

If N

N = N

But the number of oscillators, N

N

Putting these values in eqn., we get

N = N0 +N0+N0+......N0

= N0(1+++.....)

The total energy of N oscillators will be

Hence the average energy per oscillator is

(on dividing numerator and denominator bye-hv/Kt)

Thus we see that the average energy of the oscillator is not Kt (as given by classical theory)but equal to hv/(ehv/kt-1) according to Planck’s quantum theory.

Further, it can be deduced that he number of oscillators per unit volume having frequent in the range of v and v+dv is equal to

Hence the average energy per unit value (i.e., energy density) inside the enclosure is obtained by multiplying with i.e. it is given by

Putting and ,the average energy per unit volume in the enclosure of the wave lights between and +d will be

Or the energy radiated by the black body corresponding to wavelength is

Which is Planck’s radiation law or Pluck’s distribution law?

The above equation is also quite often written in the form

where and c

For more help in

0,hv,2hv,3hv....nhv

If N

_{0, }N_{1},N_{2.}...are the number of oscillators per unit volume of the hologram possessing energies0,hv,2hv....respectively, then the total number of oscillators N per unit volume will beN = N

_{0}+N_{1+}N_{2}+.....But the number of oscillators, N

_{r }having energy E_{r}is given by (Maxwell’s formula)N

_{r }= N_{0}Putting these values in eqn., we get

N = N0 +N0+N0+......N0

= N0(1+++.....)

The total energy of N oscillators will be

Hence the average energy per oscillator is

(on dividing numerator and denominator bye-hv/Kt)

Thus we see that the average energy of the oscillator is not Kt (as given by classical theory)but equal to hv/(ehv/kt-1) according to Planck’s quantum theory.

Further, it can be deduced that he number of oscillators per unit volume having frequent in the range of v and v+dv is equal to

Hence the average energy per unit value (i.e., energy density) inside the enclosure is obtained by multiplying with i.e. it is given by

Putting and ,the average energy per unit volume in the enclosure of the wave lights between and +d will be

Or the energy radiated by the black body corresponding to wavelength is

Which is Planck’s radiation law or Pluck’s distribution law?

The above equation is also quite often written in the form

where and c

_{2}=hc/k are universal constants.For more help in

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